Terramechanics

Terramechanics

TerramechanicsLand Locomotion Mechanics

Tatsuro Muro

Department of Civil and Environmental EngineeringFaculty of EngineeringEhime UniversityMatsuyama, Japan

and

Jonathan O’Brien

School of Civil and Environmental EngineeringUniversity of New South WalesSydney, Australia

A.A. BALKEMA PUBLISHERS LISSE /ABINGDON / EXTON (PA) /TOKYO

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Copyright © 2004 Swets & Zeitlinger B.V., Lisse, The Netherlands

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Published by: A.A. Balkema Publishers, a member of Swets & Zeitlinger Publisherswww.balkema.nl and www.szp.swets.nl

ISBN 90 5809 572 X (Print edition)

This edition published in the Taylor & Francis e-Library, 2005.

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ISBN 0-203-02415-X Master e-book ISBN

Contents

Preface ix

CHAPTER 1 INTRODUCTION 11.1 General 11.2 Mechanics of Soft Terrain 2

1.2.1 Physical properties of soil 31.2.2 Compressive stress and deformation characteristics 61.2.3 Shear stress and deformation characteristics 8

1.3 Mechanics of Snow Covered Terrain 191.3.1 Physical properties of snow 191.3.2 Compressive stress and deformation characteristics 221.3.3 Shear stress and deformation characteristics 26

1.4 Summary 29References 31Exercises 33

CHAPTER 2 RIGID WHEEL SYSTEMS 352.1 At Rest 36

2.1.1 Bearing capacity of weak terrain 362.1.2 Contact pressure distribution and amount of sinkage 36

2.2 At Driving State 392.2.1 Amount of slippage 392.2.2 Soil deformation 402.2.3 Force balances 452.2.4 Driving force 472.2.5 Compaction resistance 512.2.6 Effective driving force 532.2.7 Energy equilibrium 54

2.3 At Braking State 552.3.1 Amount of slippage 552.3.2 Soil deformation 562.3.3 Force balances 592.3.4 Braking force 612.3.5 Compaction resistance 652.3.6 Effective braking force 672.3.7 Energy equilibrium 67

2.4 Simulation Analysis 682.4.1 Driving state 702.4.2 Braking state 74

vi Contents


CHAPTER 3 FLEXIBLE-TIRE WHEEL SYSTEMS 833.1 Tire Structure 843.2 Static Mechanical Characteristics 863.3 Dynamic Mechanical Properties 91

3.3.1 Hard terrain 913.3.2 Soft terrain 94

3.4 Kinematic Equations of a Wheel 1093.5 Cornering Characteristics 1123.6 Distribution of Contact Pressure 1163.7 Summary 119

References 119Exercises 120

CHAPTER 4 TERRAIN-TRACK SYSTEM CONSTANTS 1234.1 Track Plate Loading Test 1244.2 Track Plate Traction Test 1244.3 Some Experimental Results 127

4.3.1 Effects of variation in grouser pitch-height ratio 1274.3.2 Results for a decomposed granite sandy terrain 1304.3.3 Studies on pavement road surfaces 1314.3.4 Scale effects and the model-track-plate test 1344.3.5 Snow covered terrain 144


CHAPTER 5 LAND LOCOMOTION MECHANICS FOR 149A RIGID-TRACK VEHICLE5.1 Rest State Analysis 149

5.1.1 Bearing capacity of a terrain 1495.1.2 Distribution of contact pressures and amounts of sinkage 150

(1) For the case where sf 0 ≥ H , sr0 ≥ H 152(2) For the case where 0 ≤ sf 0 < H < sr0 153(3) For the case where sf 0 > H > sr0 ≥ 0 154(4) For the case where sf 0 < 0 < H < sr0 155(5) For the case where sf 0 > H > 0 > sr0 156

5.2 Driving State Analysis 1585.2.1 Amount of vehicle slippage 1585.2.2 Force balance analysis 1595.2.3 Thrust analysis 162

(1) Main part of track belt 163(2) Contact part of front-idler 165(3) Contact part of rear sprocket 166

Contents vii

5.2.4 Compaction resistance 1675.2.5 Energy equilibrium equation 1705.2.6 Effective driving force 171

5.3 Braking State Analysis 1745.3.1 Amount of vehicle slippage 1745.3.2 Force balance analysis 1755.3.3 Drag 176

(1) Main part of track belt 176(2) Contact part of the front-idler 180(3) Part of rear sprocket 181

5.3.4 Compaction resistance 182(1) For the case where 0 ≥ sf 0i ≤ sr0i 182(2) For the case where sf 0i > sr0i > 0 183(3) For the case where sf 0i < 0 < H < sr0i 183(4) For the case where sf 0i > H > 0 > sr0i 183

5.3.5 Energy equilibrium analysis 1835.3.6 Effective braking force 184

5.4 Experimental Validation 1875.5 Analytical Example 197

5.5.1 Pavement road 1975.5.2 Snow covered terrain 202


CHAPTER 6 LAND LOCOMOTION MECHANICS OF 209FLEXIBLE-TRACK VEHICLES6.1 Force System and Energy Equilibrium Analysis 209

(1) During driving action 212(2) During braking action 212

6.2 Flexible Deformation of a Track Belt 2126.3 Simulation Analysis 215

6.3.1 At driving state 2186.3.2 At braking state 221

6.4 Theory of Steering Motion 2246.4.1 Thrust and steering ratio 2286.4.2 Amount of slippage in turning motion 2296.4.3 Turning resistance moment 2316.4.4 Flow chart 232

6.5 Some Experimental Study Results 2356.5.1 During self-propelling operation 2356.5.2 During tractive operations 238

6.6 Analytical Example 2386.6.1 Silty loam terrain 239

(1) Trafficability of a bulldozer running on soft terrain 239(2) Size effect of vehicle 246(3) Effect of initial track belt tension 249

viii Contents

6.6.2 Decomposed granite sandy terrain 254(1) At driving state 255(2) At braking state 259

6.6.3 Snow covered terrain 261(1) At driving state 262(2) At braking state 266


INDEX 273

Preface

Terramechanics is a field of study that deals with the physical mechanics of land locomotion.It concerns itself with the interaction problems that occur between terrain and various kindsof mobile plant. This book seeks to explain the fundamental mechanics of the vehicleterrain interaction problem as it relates to the operation of many kinds of construction andagricultural vehicles. In particular, this text seeks to clarify matters that relate to the problemof flotation, trafficability and the mobility of wheeled and tracked vehicles running on softterrains. Within terramechanics, it is clear that the ability of a construction or agriculturalmachine to do effective work in the field (i.e. its workability) depend to a very large degreeon the physical properties of the terrain and primarily upon the strength and deformationcharacteristics of the soil, snow or other material. Because of this, a number of people,who are involved in the manufacture and use of working machines, need to understand themechanical properties of ground materials like sandy or clayey soil and snow as encounteredin the construction or agricultural fields. These people include design engineers – typicallyemployed by the original equipment manufacturers, civil engineers – typically employedby the machinery users – and agricultural engineers who are commonly employed by clientand land user groups. Through the use of several practical examples, the author aims tointroduce terramechanics as a fundamental learning discipline and to establish methods fordeveloping the workability and efficiency of various types of machinery operating acrossvarious kinds of soft terrain.

The author also seeks to build up a design method for a new, rational, processfor working systems design that can be applied by civil, mechanical and agriculturalengineers engaged in day to day field operations. Alternately, the process methodol-ogy may be employed by graduate and undergraduate students of terramechanics todevelop sound principles of analysis and design within their study programs. In termsof historical background, the modern academic discipline of terramechanics and land loco-motion mechanics may be considered to have been ‘invented’ by M.G. Bekker over aperiod of 20 years in the 1950s and 60s. Bekker’s ideas were expounded in three classicworks entitled ‘Theory of Land Locomotion’ (1956), ‘Off-the-road Locomotion’ (1960)and ‘Introduction to Terrain-vehicle Systems’ (1969). These works are now considered tobe the original ‘bibles’ in this field. Following on from Bekker, many early studies relatingto the performance of wheeled and tracked vehicle systems took an essentially empiricalapproach to the subject. In recent times, however, academic studies in this field, and inassociated fields such as soil and snow mechanics, have evolved into highly sophisticateddomains involving both theoretical mechanics as well as experimental truths. As a conse-quence, modern land-locomotion mechanics may be considered to be approaching a fullmathematical and mechanical science.

In relation to this newly developing science, a systematic approach to the layout of thistext has been adopted. Also, in the body of this work, the importance of properly conducted

x Preface

experiments to determine the soil-machine system constants, the significance of the sizeeffect, the problem of the interaction between wheel, track belt and terrain and the validityof the simulation analysis method is stressed. In Chapter 1, the mechanical properties e.g.the compressive and shear deformation characteristics of soil materials in soft terrain andsnow materials in snow covered terrain are analysed. Also, several kinds of test methodwhich are used to judge correctly the bearing capacity and the trafficability of wheeled andtracked vehicle are explained in detail. In Chapters 2 and 3, the land locomotion mechanicsof wheeled vehicle systems composed of both rigid wheel and flexibly tired wheels areanalysed. New simulation analytical methods are also introduced that use relations betweensoil deformation, amount of slippage at driving or braking state, the force and momentbalances among driving or braking force, compaction resistance and effective driving orbraking force to predict performance. Following this, several analytical and experimentalexamples are given.

In Chapter 4, some terrain-track system constants that may be used to properly evaluatethe problem of interaction between track belt and terrain materials are developed fromtrack plate loading and traction test results. Following this, the most important problemsof determining the optimum shape of track plate to develop maximum tractive effort andthe size effect of track plates on a soft terrain are analysed in detail from the perspective ofseveral experimental examples.

In Chapters 5 and 6, the land locomotion mechanics of a tracked vehicle equipped withrigid and flexible track belts are analysed for the case of straight-line forward and turningmotions on a soft, sloped, terrain. In relation to the traffic performances of both the rigidtracked vehicle and flexible tracked vehicle, an analytical method is used to obtain therelationship that exist between driving or braking force, compaction resistance, effectivedriving or braking force and slip ratio or skid, and contact pressure distribution. Theserelations are developed on the basis of the amounts of slippage and sinkage of track belt,energy equilibrium equation and the balance equation of the various forces acting on thetracked vehicle during driving or braking action. Following this, a new simulation analyticalmethod is employed to calculate the amount of deflection of the flexible track belt and thetrack belt tension distribution during driving or braking action. Subsequent to this, severalanalytical examples of the traffic performances of a flexible tracked vehicle running onpavement road, decomposed weathered granite sandy terrain, soft silty loam terrain andsnow covered terrain are presented and discussed.

In conclusion, it is observed that in recent years there has been a remarkable growth ininterest in the potential use of robotized and un-manned working vehicles in the field ofconstruction and agricultural machinery. The emerging academic field of terramechanicshas great potential value here since it can give important suggestions to designers fordevelopment of the workability and the trafficability of potential robotized machineryintended for operation on soft earth and other forms of soft terrain.

To conclude this preface, it is the sincere hope of this author that readers will find thiswork to be useful. It is hoped that the book will give reliable guidance and informationto students, mechanical engineering designers, robotic-machinery developers and to fieldengineers. In the writing of this book, the author has obviously sought to explain the basictheoretical mechanics as clearly as he can but nevertheless he is aware that there maybe many deficiencies in layout and exposition. For improvement, reader suggestions andfeedback and are earnestly sought. In the production of a book many people are involved.

Preface xi

The author wants to express his particular and most sincere thanks to Mr. H. Mori of theDepartment of business, to Mr. T. Ebihara and Mrs. Y. Kurosaki of the editorial departmentof Gihoudo Press, and to Mr. K. Kohno of Ehime University.

To my Australian collaborator, Jonathan O’Brien of the University of New South Wales,I give special thanks for his very high quality final English translation, for his technicalediting and for his many suggestions and constructive criticisms. Further, I would speciallylike to thank him for the very large amount of work done in relation to the production of thecamera-ready version of the book. However, the primary responsibility for the technicalcontent of this work and for any errors or omissions therein remains that of the seniorauthor.

Tatsuro MuroMatsuyama, Japan

Chapter 1

Introduction

1.1 GENERAL

Over the last century, the mobile construction and agricultural equipment domain has risenfrom a position of virtual non-existence to one of major industrial significance. Nowadays,for example, everyone is familiar with heavy construction equipment that utilise eithertracked or wheeled means for the development of powered mobility. The collective technicalterm used to refer to the mobility means used to propel and manoeuvre machines overvarying terrains is ‘running gear’. Wheels and tracks are the predominant types of runninggear used in the mobile equipment industry but they are not the only ones. Legged machinesand screw propulsion machines, for example, can be utilised – but they are quite rare.

Given the industrial importance of mobile machinery, a field of study that analysesthe dynamic relationship between running gear and the operating terrain is of consequentmajor importance. The modern field of study that addresses this subject area is called ‘Terra-mechanics’. The output of this field of study is qualitative and quantitative information thatrelates to such questions as the off-road trafficability of a vehicle, its travel capacity andthe bearing capacity of a particular piece of ground. This type of information is necessaryfor the design of mobile machines and for their continuous performance improvement.

A central interest in terramechanics is that, relatively shallow, piece of ground that liesunder a vehicle and which can become directly involved in a dynamic interplay with theequipment’s running gear. The mechanical and deformation properties of this piece ofterrain under both compressional and shear loading are defining in being able to predict thecomposite vehicle terrain behaviour. Many studies have been carried out to elucidate thecomplex relationship between the very many characteristics of the terrain and those ofthe vehicle. The design of the dimension and shape of tracks and wheels is very importantwhen vehicles have to operate upon and across natural surfaces. This is so that vehicles whenthey are on weak terrain or snow do not suffer from excessive amount of sinkage when thebearing capacity of the ground underneath them is exceeded. Overcoming the sinkageproblem requires a study of the phenomenon of vehicle ‘flotation’. The study of this factornecessitates the study of the bearing capacity of the ground when the vehicle is both at restand when it is in powered or unpowered motion.

Consideration of the vehicle in motion is required, since when the vehicle is in a self-propelling, driving or braking state, the wheel (or track) must effectively yield the shearresistance of the soil so as to generate the necessary thrust or drag. To ensure these generatedproducts, a proper design of the grouser shape of the track and an appropriate selectionof pressure distribution as well as the tread pattern and axle load of the tire is required.Further, it is necessary to study empirical results and field measurements if one wishesto maximize ‘effective draw-bar pull’ and/or to determine the forces required to overcome

2 Terramechanics

various motion resistances. These resistances include the slope resistance (which occurswhen the vehicle is running up slopes) and compaction resistance (which is a consequenceof the rut produced by static and slip sinkage).

The flotation, off-the-road trafficability and working capacity of a vehicle all dependprincipally on ground properties. In general, the ground property relating to the running ofa vehicle is called ‘Trafficability’.

On the other hand, the mobility of a piece of construction machinery depends on a hostof vehicle parameters. These include factors that relate to engine power, weight of vehicle,spatial location of center of gravity, width and diameter of wheel, shape of tire tread, widthand contact length of track, initial track belt tension, application point of effective draw-barpull i.e. effective tractive effort or effective braking force, mean contact pressure, shape,pitch and height of grouser, diameter of front idler and rear sprocket, number of road rollers,suspension apparatus, type of connection and minimum clearance. In general, the runningcapability of a vehicle is called its ‘mobility’.

The difference between the notions of ‘trafficability’and ‘mobility’ is somewhat circular.‘Trafficability’ may be thought of the ability (or property) of a section of terrain to supportmobility. ‘Mobility’ is defined as the efficiency with which a particular vehicle can travelfrom one point to another across a section of terrain. To a degree, trafficability is a terraincharacteristics centered concept whilst mobility is a task and vehicle-configuration centeredconcept.

Studies of both trafficability and mobility can provide a very useful background tothe design and development of new construction machinery systems that aim to improveworking capacity. These studies are most useful to those who wish to select a best or mostsuitable machine for a given terrain.

Now, let us turn to consideration of a number of experimental procedures that may beused to measure and to generally investigate the mechanical properties of the terrain surfaceof weak soil and of snow covered terrain. These experimental procedures can serve as thebasis for a thorough study of the mobility characteristics of vehicles which may operate ineither a driving or in a braking state. The methods also allow study of the turnability oftracked vehicles and the cornering properties of wheeled vehicles. Further, these approachespermit study of the travelling features of vehicles on slopes and allow study of the travellingmechanics of rigid or flexibly tracked vehicles or wheeled systems.

1.2 MECHANICS OF SOFT TERRAIN

In this section we will review some experimental field procedures that may be carried out todetermine the soil constants for a particular terrain and which at the same time can help todefine some of the physical properties, the shear resistance deformation characteristics, andthe compression deformation characteristics of the soil. Knowledge of these soil propertiesis necessary to determine the trafficability viz. the thrust or the drag and the soil bearingcapacity of tracked and wheeled vehicles which might be operating in a driving or brakingstate. The vehicles may be running on weak clayey soil or on loosely accumulated sandy soil.

In terms of available tests, use of the cone index to determine trafficability is of particularinterest. The cone index will be discussed in more detail in later sections.

Introduction 3

Figure 1.1. Idealised three-phase soil system.

1.2.1 Physical properties of soil

Natural soil typically consists of solid soil particles, liquid water and air in the void space.Figure 1.1 shows the typical structure of a moist natural soil in volumetric and gravimetricterms. The soil can also be thought of a being divided into two parts: a saturated soil partcontaining pore water and an unsaturated soil part.

The specific gravity of the solid soil particles may be measured according to the proce-dures of JIS A 1202 (say) by use of a pycnometer above 50 ml. The result is usually in therange from 2.65 to 2.80. If the volume of the soil particles is Vs (m3) their weight is Ws

(kN), the void volume is Vv (m3), the volume of pore water is Vm (m3), the weight of porewater is W (kN), the total volume of the soil is V (m3) and the total weight of the soil is W(kN), then the unit weight γ is given by:

γ = W

V= Ws + Ww

V= Gγw (kN/m3) (1.1)

where G is the apparent specific gravity of the soil and γw is the unit weight of water. Thevoid ratio e and the porosity of the soil n are non-dimensional and are defined as:

e = Vv

Vs(1.2)

n = Vv

V(1.3)

The water content w and the degree of saturation Sr are non-dimensional and are defined as:

w = Ww

Ws× 100 (%) (1.4)

Sr = Vw

Vv× 100 (%) (1.5)

In the case of saturated soil:

e = wGs (1.6)

For the case of an unsaturated soil, the following equation may be developed:

G = Gs(1 + w)

1 + e(1.7)

4 Terramechanics

Similarly formulae for the wet density ρt and the dry density ρd may be developed asfollows:

ρt = W

gV(g/cm3) (1.8)

ρd = Ws

gV= ρt

1 + w/100(g/cm3) (1.9)

where g is the gravitational acceleration.The relative density of a soil Dr can be expressed by the following equation where emax

is the maximum void ratio when the soil is loosely filled, emin is the minimum void ratiowhen the soil is most compacted and e is the natural void ratio.

Dr = emax − e

emax − emin(1.10)

The value of the relative density for a loosely accumulated sandy soil lies in a range between0 and 0.33.

The grain size analysis recommendations of national and international testing standards(such as JIS A 1204), indicate that the size of soil particles of diameter larger than 74µ canbe determined by simple mechanical sieve analysis. Particles of diameters less than 74µ

can be determined by sedimentation analysis using the hydrometer method.Figure 1.2 shows an experimentally derived grain size distribution curve for a typical

soil. The abscissa represents the diameter of soil particle on a logarithmic scale whilst theordinate represents the percentages finer by weight. If D60, D30 and D10 are diameters ofthe soil particles that have a percentage finer by weight of 60%, 30% and 10% respectively,then the coefficient of uniformity Uc and the coefficient of curvature U ′

c are respectively:

Uc = D60

D10(1.11)

U ′c = D2

30

D60 × D10(1.12)

Figure 1.2. Particle size distribution curve.

Introduction 5

In general, if the grain size distribution curve of a soil has a Uc is greater than 10 and ifU ′

c is in the range of 1 ∼ 3 and shows a S shape curve, then the soil is considered to have agood distribution [1].

Table 1.1 gives a classification of gravel, sand, silt and clay according to the Japaneseunified soil classification system [2]. Alternatively, where one has determined the percent-age of sand, silt and clay in a soil based on its grain size distribution curve, the name ofthat soil can be presented by use of a triangular soil classification system [3].

The angle of internal friction of a sandy soil depends mainly on its particle properties.However, it also depends on the shape of the individual soil particles and their particularsurface roughnesses.

Figure 1.3 shows a typical sectional form of a soil particle. The slenderness ratio and themodified roundness of the sectional form [4] are defined as:

Slenderness ratio = a

b(1.13)

Modified roundness = 1

2

(r2 + r4

a+ r1 + r3

b

)(1.14)

where a and b are the apparent longer and shorter axes and r1 ∼ r4 are the radii of curvatureof the edge parts of the soil particle.

Table 1.1. Principal particle size scale [2].

1�m 5 �m 74 �m 0.42 mm 2.0 mm 5.0 mm 20 mm 75 mm 30 cm

Colloid

ClaySand

Silt

Fine sand Coarse sand Coarse gravelFine gravel Medium gravel

GravelCobble Boulder

Fine particle Coarse particleRock materials

Soil materials

Figure 1.3. Measurement of slenderness ratio and modified roundness.

6 Terramechanics

For groups of particles, these values should be expressed as an average. This is because (ofnecessity) the values need be determined from measurements taken from many individualparticles. Measurement of group properties may be obtained by a sampling process whereinmany representative particles are obtained by coating a microscope slide with a low cohesionglue. If the sticky slide is placed in the soil, thin layers of random particles will adhere tothe slide. In scientific sampling work, slides need to be taken both in the horizontal and inthe vertical plane of the soil.

In contrast to internal frictional values, the shear deformation characteristics of a cohesivesoil depend largely on its water content. Thus, if we remove water from a cohesive soil inliquid state, it will transform into a more plastic state with an increase of cohesiveness.Further, with continued water content decrease it will become semi-solid and then solid.A measure of this physical metamorphism is called the ‘Consistency’ of the clay. Also, thelimit of the water content is called the liquid limit wL, the plastic limit wp and the shrinkagelimit ws. These soil mechanics parameters are defined in many international and nationalstandards. In Japan, the appropriate standards are JIA A 1205, 1206 and 1209.

The range of water contents over which a specimen exhibits plasticity can be expressedby a plastic index Ip, which is defined as:

Ip = wL − wp (1.15)

Similarly, a consistency index Ic that shows the hardness and the viscosity of fine grainedsoil may be defined as:

Ic = wL − w

Ip(1.16)

When Ic approaches zero, the soil tends to be changed easily to a liquid state by disturbancefrom vehicular traffic. However, when Ic approaches one, the soil is a stable one becausethe soil is in an over-consolidated state with lower water content.

1.2.2 Compressive stress and deformation characteristics

A natural ground surface operated upon by a surface vehicle will typically be compressedand deformed as a result of the weight of the machinery passing into the soil through itswheels or tracks. Under rest conditions, the bearing characteristics of soil under the trackof the construction machine, for example, can be calculated by the use of Terzaghi’s limitbearing capacity equation [5,6]. This equation uses a coefficient of bearing capacity basedupon the cohesion of the soil and its angle of internal friction. However, this equation cannot be used to calculate the static amount of sinkage which is necessary to predict thetrafficability of vehicles. Also, Boussinesq’s method [7] for prediction of the soil stress inthe ground based on the load on the semi-infinite elastic soil does not predict the staticamount of sinkage under conditions where plastic deformation occurs – such as in weakterrains and in loosely accumulated sandy soils.

In contrast, the experimental plate loading test – as defined by JIS A 1215 – may beused as a direct field measurement of the coefficient of subgrade reaction of a section ofroad or base course. Using this method, direct measurement of the load sinkage curve ispossible – provided that a standard steel circular plate of 30 cm diameter and thicknessmore than 25 mm is utilised.

Introduction 7

In general, because the size of the loading plate has an effect on the amount of sinkagemeasured, it is necessary to develop some adjustment formula to allow correlation to theaction of specifically sized wheels and tracks. More specifically, the following equation[8] can be employed to adjust for different contact areas of wheel and track. In the case ofsandy soil, the amount of sinkage s (cm) of track of width B (m) may be calculated by useof the following formula.

s = s30

(2B

b + 0.30

)2

(1.17)

where s30 (cm) is the amount of sinkage of a standard 30 × 30 cm2 loading plate and theother part of the expression is a size adjustment factor. In the case of clayey soil, theamount of sinkage s (cm) for a track of width B (m) increases with an increase of B. It canbe calculated for the same loading plate through use of the following formula [9].

s = s30B

0.30(1.18)

To relate the amount of sinkage s0 (m) and the contact pressure p = W /ab (kN/m2) causedby a load W (kN) exerted on an a × b (a ≥ b) (m2) plate, Bekker [10] proposed the followingequation (modifying the Bernstein-Goriatchkin’s equation) which introduced the index n.Bekker suggested that the amount of sinkage increases with the width of the wheel or trackaccording to the expression:

p = ks0n =

(kc

b+ kφ

)s0

n (1.19)

where n is a sinkage index (which varies depending on the type of soil) and kc (kN/mn+1)depends on the cohesion c, and kφ (kN/mn+2) depends on the angle of internal friction φ.The value of these sinkage coefficients can be determined by experiments using plates oftwo different b values [11].

The above equation – with size effect taken into consideration – reduces to Taylor’sequation [12] when n = 1.

To obtain the values of the constants k and n, Wong [13] proposed the following Ffunction (based on a least squares analysis method) which gives a weighting factor to thecontact pressure p. This method takes as a basic assumption the idea that all observationshave an equal reliability to error.

F =∑

p2 (ln p − ln k − n ln s0)2 (1.20)

More precise calculations of k and n can be achieved by solving the two expressions derivedfrom the above equation when the partial differentials ∂F/∂k and ∂F/∂n are set to zero.

Taking a somewhat different approach, Reece [14] described the relationship betweenthe amount of static sinkage s0 (m) and the contact pressure p (kN/m2) by the followingequation.

p = (ck ′c + γbk ′

φ)( s0

b

)n(1.21)

where c (kN/m2 ) is the cohesion, γ (kN/m3) is the unit weight, b (m) is the shorter lengthof the rectangular test-plate and k ′

c and k ′φ are non-dimensional soil constants.

8 Terramechanics

To complicate matters further, in the case when these equations are applied to actual tracksor wheels as loading devices the rectangular plate should be modified to correspond to thetire tread pattern or the grouser shape of the track. Furthermore, additional experimentsmust be executed to allow for inclined load effects under traction. Yet again, in cases whereimpact is applied to a relatively hard clayey soil or in cases where there exists punchingaction on a compound soil stratum, some different considerations apply.

To elucidate these matters, some experimental data for a model tracked-vehicle willbe presented in Chapter 4. This work will discuss experimental determinations of theterrain-track constants from the plate loading test.

1.2.3 Shear stress and deformation characteristics

When a track or wheel is running free but is acted on by a positive driving or negativebraking action and/or when the track or wheel acts upon a weak terrain, it experiences aphenomenon called ‘Slip’. Slip involves relative motion of the running element and thesoil and leads to the development of shear resistance due to the shear deformation of thesoil. As a consequence, the soil exerts a thrust or drag corresponding to the amount ofslippage necessary for the running of the vehicle. In addition, some degree of slip sinkageof the track or wheel will occur at the same time. This sinkage arises from a bulldozingtype of action that develops as well to a volume change phenomenon that occur in the soilconsequent to its shear deformation.

Let us now review some experimental methods and processes that can be used to deter-mine the stress-strain curve and the volume change relation with strain for a soil. These tworelations characterize the shear deformation relationship that applies to a particular soil.

(1) Direct shear testFigure 1.4 shows a shear box apparatus comprised of two disconnected upper and lowerboxes. The box can be filled with a soil sample at a certain density. If a static normal load Pis applied to the specimen through a platen and the lower box is moved laterally a shearingaction in the horizontal direction results. This action occurs over a shearing area A. Therelationship between the volume change of the specimen (typically a small contraction ordilatation) and the shear deformation of the soil sample can be determined experimentallyas can the relationship between the shear force T and the horizontal deformation of thelower box. This experimental method is formally referred to as the ‘box shear test’ [15].

Figure 1.4. Direct shear box test apparatus.

Introduction 9

In general, the force P is exerted through a gravity operated lever. The force T is typicallymeasured by use of a proving ring equipped with an internal dial gauge. Measurement of thehorizontal displacement of the lower box and the vertical displacement of the loading platenecessary for the measurement of the volume change of the soil samples can be carried outby the use of two dial gauges. The experiment is typically carried out at a shearing speedof between 0.05 ∼ 2 mm/min.

Figure 1.5 shows the general pattern of results that is achieved using this apparatus. In thediagram the relationship between shear resistance τ = (T /A) and horizontal displacementas well as the relationship between the volume change and the horizontal displacement isshown for conditions of constant normal stress σ = (P/A). Typically, two types of experi-mental results are obtained for what one can refer to as type A and type B soils. In type Asoils, the phenomenon is one where change of shear resistance τ and the contracting volumechange tend towards a constant value or asymptote. This behaviour is typical of normallyconsolidated clays and loosely accumulated sandy soils. In contrast, the shear resistance τ

of type B soils typically shows a marked peak at a certain horizontal displacement. Also, theexpansive volume change occurs with a dilatancy phase following an initial contractionalvolume change. These hump type behaviours can be usually observed in overconsolidatedclays and in compacted sandy soils. From these curves, a determination of the cohesionc and the angle of internal friction φ can be made through the use of Coulomb’s failurecriterion [16]. The process by which this is done is to repeat the shear test for a numberof normal force values P, to record the peak shear resistance and then to plot the resultingdata on a graph.

Figure 1.5. Relationship between shear strength, volume change and horizontal displacement.

10 Terramechanics

Figure 1.6. Determination of cohesive strength c and the angle of internal friction φ.

Figure 1.6 plots the peak shear resistance τmax for a B type soil against the normal stressσ. The shear resistance τ value is that measured at a horizontal displacement of 8 mm (or50% of the initial thickness of the soil sample) for an A type soil.

τ = c + σ tan φ (1.22)

The importance of this test can be appreciated from the fact that the bearing capacity ofa terrain as well as the maximum thrust and the maximum drag of a tracked or wheeledvehicle system can be calculated using the cohesion c and the angle of internal friction φ.

One problem that arises here though, concerns the fact that the slip velocity of vehiclesin practice may be greater than 1 m/s. This is an order of magnitude greater than that used inshear box test rate which is around 0.05 mm/min. This indicates that use of the latter shearrate to describe the actual shear deformation characteristics occurring under the wheels ofreal vehicle systems is clearly not appropriate.

In an attempt to overcome this loading rate difficulty, Kondo et al. [17] in 1986 developeda high speed shearing test for cohesive soils using a dynamic box shear test apparatus basedon the principle of Hopkinson’s bar. Their method, however, has not come into widespreaduse. A yet further deficiency of the standard box shear test, relates to the fact that thetotal shear displacement is much too small in comparison to the actual amount of slippageexperienced with real tracked or wheeled vehicles. The arguments indicate why directapplication of the relationships measured between shear resistance τ, volume change andhorizontal displacement in the shear box to real world vehicle systems is not appropriate.Further, the shear box test does not include any of the bulldozing effects that typically occurin front of wheel systems. These methodological limitations makes effective prediction ofthe amount of slip sinkage that is associated with the slip of track or wheel impossibleby use of this box shear method. To overcome some of these difficulties Chapter 4 willintroduce an experimental procedure that is based upon a model tracked-vehicle. Thisexperimental method is necessary to determine real value terrain-track system constantsand is indispensable to the calculation of thrust and drag values as well as for calculationof the amount of slip sinkage that occurs under powered vehicles.

(2) Unconfined compression testThe unconfined compression test is a test that is very commonly encountered in generalmaterials testing and in soil mechanics. For example, JISA 1216 describes the experimentalprocedures for carrying out an unconfined compression test in a standard manner. Using this

Introduction 11

Figure 1.7. Relation between compressive stress σ and compressive strain ε.

Figure 1.8. Mohr’s failure circle.

procedure or similar ones (such are as prescribed by the ASTM or other testing authorities)the compressive stress–strain curve and the unconfined compressive strength qu of a sampleof a cohesive soil can be easily obtained. In the standard form of test, the relationshipbetween compressive stress σ and compressive strain ε is measured at a compressive strainrate of 1%/min. The standard compression test specimen is a soil sample of cylindricalform. The specimen diameter is standardised at 35 mm and the height is set at 80 mm.A typical stress–strain curve for a test on a cohesive soil sample is shown in Figure 1.7.

This method is obviously not appropriate to the testing of sands and other granular solids.The stress developed in the stress–strain curve at the point where ε reaches 15% is deemed tobe the unconfined compressive strength qu. A value of a modulus of deformation ε50 can bedeveloped based on the inclination of a line which links the origin and a point correspondingto the deemed compressive strength qu/2 on the stress–strain curve. A Mohr’s failure stresscircle for this case is shown in Figure 1.8. In the case of a saturated clayey soil, qu/2 equalsthe undrained shear strength of the soil cu.

Further to this, it is an empirical observation from the field that, in general, when atrack or wheel shears a saturated clayey soil at high speed, the undrained shear strengthis constant and does not depend on the confining pressure. This result develops becauseshearing action is taking place under undrained conditions – as a consequence of pore waterdrainage not occurring.

As a consequence of this, the undrained shear strength cu is very important in makingassessments as to the trafficability of vehicles over given terrains.

12 Terramechanics

Figure 1.9. Vane shear test apparatus.

A sensitivity ratio St which expresses the sensitivity of a particular clayey soil to disturbancefrom passing vehicles can be developed using the following formula:

St = qu

qur(1.23)

where qur is the unconfined compressive strength of the undisturbed clayey soil and qur

is the unconfined compressive strength when it has been remolded with a water contentidentical to that of the undisturbed soil.

The value St of a cohesive soil typically lies in the range 2 ∼ 4. A cohesive soil with St > 4is called a sensitive clay. The degree of sensitivity of a clayey soil also plays an importantrole in assessments of the trafficability of a terrain to vehicles.

(3) Vane shear testThe vane shear test method is the most normal procedure employed for in-situ measurementof very soft or weak cohesive terrains. Figure 1.9 shows the key features of the vane testapparatus. The vane itself is comprised of four blades which are rectangular plates of heighth and width d/2 intersecting at right angles to each other. Shearing action in a clayey soiloccurs when the vane is rotated continuously after its penetration into the ground. Therelationship between the cohesion c and the maximum shearing torque Mmax necessaryfor the vane to shear cylindrically a particular clayey soil in a cylindrical fashion can beexpressed through use of the following formula – providing that the frictional resistancebetween the soil and the vane axle is ignored.

c = Mmax

π (hd2/2 + d3/6)(1.24)

In this formula the blade diameter d is usually 5 cm and the shape is set as h = 2d.Generally, the shear strength of a clayey soil shows anisotropy in different directions.

Indeed, there are cases where the shear strengths in the horizontal and vertical directionsare substantially different. Although the phenomenon of soil shearing by a track belt (suchas with a tracked vehicle) typically occurs in a horizontal direction, the shear strengthobtained in the vane shear test mainly takes place in the vertical direction. To determinethe horizontal shear strength τH , Bjerrum’s equation [18] can be applied. He found that theratio of the horizontal shear strength τH to the vertical one τV is in inverse proportion to

Introduction 13

the plastic index Ip. In the general case, the shear strength τβ in an arbitrary direction β tothe horizontal plane can be calculated using Richardson’s equation

τβ = τH τV(τ2

H sin2β + τ2V cos2β

)− 12 (1.25)

Shibata [20] pointed out that the ratio of anisotropy τV /τH for vane shear strength increasesremarkably with decreases in the plasticity of the cohesive soil. He also suggested that, forcohesive soils of plasticity index of around 10, the ratio is almost equal to the coefficientof earth pressure at rest.

A further problem in relation to real situations occurs because the tractive characteristicsof track-laying vehicles running on a very soft ground (i.e. the thrust or the amount ofsinkage) are greatly influenced by the phenomenon of increase of shear strength with depth.Typically, ground shear strength develops linearly with depth. This problem can be analysedby using large deformation elasto-plastic FEM [21] (finite element analytical methods) inwhich the change of the undrained shear strength and the elastic modulus in the directionof depth is measured by means of the vane shear test. Moreover, in relation to the effect ofshear rate on vane shear strength τV ,Yonezu [22] and Umeda [23] carried out an experimentwhere the rotation speed of the vane was in the range of ω = 1.75×10−3 ∼ 1.62×10−1 rad/sto 8.73 × 10−4 ∼ 8.73 × 10−3 rad/s. They showed that the relation between τV and ω couldbe expressed as:

τV = a log ω + b (1.26)

In this formula, a is a coefficient which relates to the type of cohesive soil and b is a valuewhich varies depending on depth. This type of research is extremely important in the fieldof terramechanics as it assists in the development of a clear understanding of the rate effecton the shear resistance of cohesive soils under the action of tracks or wheels. This topic,however, still requires much further investigation and research.

(4) Bevameter testFigure 1.10 shows a Bevameter test apparatus in which the loading plate may take the formof the grouser shape of a track or of the tire tread pattern of a wheel. Through application of anormal load through the plate, the Bevameter can be used to measure the amount of sinkageand can be used to obtain the pressure sinkage curve which characterizes the compressivestress and deformation relationship of a terrain. Thence, by applying a torque under aconstant normal load, the Bevameter can be used to measure the relationships that existbetween shear resistance, normal stress and the amount of slippage. Alternately, it maybe employed to determine the relationships that exist between the amount of slip sinkage,the normal stress and the amount of slippage which characterise the shear deformation ofthe terrain. This experimental apparatus was first developed by Söhne [24] who used it toobtain the pressure sinkage curve by employment of an hydraulic loading cylinder mountedon a vehicle.

Following this, Tanaka [25] revised the method through employment of a cone or a ringloading plate. He also developed an automatic recording device to record the normal pres-sure and the shear resistance relations. Bekker [26] developed a test vehicle equipped withplate load test apparatus which he used to obtain the pressure sinkage curve. The test vehiclewas also equipped with a Bevameter test apparatus with a ring shear head. This was usedto obtain the shear resistance deformation curve for a constant load. In 1981, Golob [27]

14 Terramechanics

Figure 1.10. Bevameter test apparatus.

developed an apparatus which combined the plate loading test and the Bevameter test. Thistest apparatus can not only measure the pressure sinkage relations and the relationshipsthat exist between shear resistance, normal stress and amount of slippage, but it can alsomeasure the relationship between the amount of slippage, normal stress and amount ofslippage.

The results of plate loading tests for plain plates or plates armed with grousers havealready been mentioned in the Section 1.2.2. Let us now consider the case where a constantnormal load P is applied to a ring head which is fitted with six vanes to simulate trackgrousers having certain grouser pitch and height ratio. In this case, the outer and innerdiameters of the rings are defined as 2ro and 2ri respectively. A relationship betweenthe shearing torque M and the displacement rotation j for a cohesive terrain can then becalculated by use of the following equation.

M = Mmax{1 − exp(−aj)}(1.27)

j = ri + ro

2α

where Mmax is the maximum shear torque, a is the shape coefficient of the shear torquecurve and α is the angle of rotation. So far as shear resistance τ is concerned, the shearingtorque M may be calculated by the following equation:

M = τ

∫ ro

ri

{π(r + dr)2 − πr2} rdr = 2

3πτ(r3

o − r3i ) (1.28)

This expression shows that the shearing torque M and the shear resistance τ have aproportional relation one to each other.

Introduction 15

Figure 1.11. Failure curve.

If τmax is the maximum shear resistance, then the relationship between the shear resistance τ

and the rotational displacement of the Bevameter j for a cohesive terrain can be expressed as:

τ = τmax{1 − exp(−aj)} (1.29)

Again, if the base area of the ring head is denoted as A, then the shear resistance–deformationcurve which occurs under various applied normal stresses σ = (P/A) can be determined.

In general, the associated amount of slip sinkage ss – which is measured at the sametime – can be expressed as a function of the normal stress σ and the amount of slippage ofsoil js. The relation can be expressed as:

ss = c0σc1 jc2

s (1.30)

where c0, c1 and c2 are constants that relate to grouser pitch and height ratio and to thetype of soil encountered. In this case js = j. The values for these particular constants canbe obtained by micro-computer analysis of the results of the plate loading test and the ringshear test through application of the least squares method. An automatic data processingprocedure for this technique has been developed by Wong [28].

(5) Triaxial compression testWith regard to the soil ingredients that comprise natural ground materials and within thegeneral domain of soil mechanics, a number of constitutive equations [29] relating materialstress and failure criteria [30] have been proposed.

Figure 1.11 shows the Von-Mises, Tresca and Mohr-Coulomb’s failure criteria expressedin a coordinate system comprised of the three principal stresses σ1, σ2 and σ3. The criteriathemselves are as given in the following expressions.[Tresca’s criterion]

τmax = 1

2(σmax − σmin) = constant (1.31)

[Von-Mises’s criterion]

τoct = 1

3

√(σ1 − σ2)2 + (σ2 − σ3)2 + (σ3 − σ1)2 = constant (1.32)

where τoct is the octahedral shear stress.

16 Terramechanics

Figure 1.12. Mohr-Coulomb failure criteria.

Figure 1.13. Triaxial compression test apparatus.

[Mohr-Coulomb’s criterion]

σmax − σmin = 2c cos φ + (σmax + σmin) sin φ (1.33)

τ = c + σ tan φ (1.34)

Most commonly, the well-known Mohr-Coulomb failure criterion is used to show the rela-tionship between the shear resistance τ and the normal stress σ, as is shown in Figure 1.12.The initial stress condition in the ground can be defined by a Mohr’s stress circle witha diameter of AK. When a tracked or wheeled vehicle traverses the ground, the stresscondition in the soil turns into an active or passive stress state.

These stress conditions can be represented by Mohr’s stress circles that have diametersAB and AC respectively. The active or passive stress states which occur on the slip failureplane can be represented by Ta or Tp respectively in Figure 1.12. From the connectingline between Ta and Tp, (i.e. the failure envelope) the cohesion c and the angle of internalfriction φ can be determined.

Figure 1.13 shows the essential features of the triaxial compression test apparatus. In theuse of this apparatus, a sample of a clayey soil is formed into a cylindrical shape by meansof a wire saw. The specimen is then sealed by covering its sides with a rubber membrane.The sample is then placed on a porous stone which is located at the base of the test apparatusand topped with a cap. The ends are then tightened towards each other.

Introduction 17

Figure 1.14. Relationship between strain rate and shear strength ratio Rt .

To determine the shear strength of a soil sample under actual drainage conditions, thesample is compressed with a vertical normal pressure σ1 whilst the sample is confinedlaterally with a constant horizontal confining pressure σ3 which is applied by hydraulicpressure. In this test arrangement, the relationship between the angle of slip failure planeα, the normal pressure σ and the shear stress τ can be expressed by the following equation.

σ = σ1 + σ3

2+ σ1 − σ3

2cos 2α (1.35)

τ = σ1 − σ3

2sin 2α (1.36)

Using these relations, the Mohr’s stress circle can be determined as:(σ − σ1 + σ3

2

)2

+ τ2 =(

σ1 − σ3

2

)2

(1.37)

In the triaxial compression test, depending on the drainage conditions utilised the experi-mental method can be classed into three different test regimes – namely the unconsolidatedundrained shear test (UU Test), the consolidated undrained shear test (CU Test) and thedrainage shear test (D Test). In the field of terramechanics, the undrained shear test (UUTest) is commonly used because of the large strain rates encountered and the undrainedcondition typically encountered in the field. Although the standard triaxial compressiontest typically uses a quite fast strain rate of 1%/min, the required strain rate of the soil interramechanics work is very much higher.

This is because there is a rapid shearing action which accompanies the dynamic loadingof a ground by a track or a wheel. As a result, since the shear strength of a soil is largelydependent on the shear strain rate the actual strength of a material in the field will typicallybe quite a lot higher than that measured by the triaxial test.

Figure 1.14 illustrates the relationship [31] that might exist between strain rate and theratio of shear strength to standard shear strength for both a clayey soil and a sandy soilsample – assuming a standard strain rate of 1%/min.

(6) Cone penetration testFigure 1.15 shows a portable cone penetrometer apparatus to which is attached a cone ofbase area A and apex angle α. When the cone is forced into a ground with a speed of1 cm/s the penetration force P (which varies with the penetration depth) can be measured

18 Terramechanics

Figure 1.15. Cone penetrometer.

by use of a proving ring. This common piece of field instrumentation is called the ‘conepenetrometer’. It is in world-wide use for assessing the trafficability of agricultural andconstruction terrains and machinery mobility. The penetration pressure acting on the baseof the cone under constant penetration speed i.e. P/A is defined as the cone index CI. Forcohesive soils, the following relation can be established between the cohesion c and thesensitivity ratio St [32].

CI = P

A= 3π

2

(1 + 1√

St

)c (1.38)

Although the apex angle α of test cones may take on values of π/6, π/3 and π/2 rad, it is anexperimental finding that different values of this angle do not have a significant influenceon the measured value of CI. However, because side friction can have a bearing on themeasured result, systems have been developed that use a double tube.

Also, load cell based systems are available that can directly monitor, record and plotpenetration force as a function of depth.

Hata [33] has analysed theoretically the relationship between the cone index CI and thecohesion c for both the initial stage and the steady state, taking into account the apex angleand the smoothness of the surface of cone. This analysis was carried out for the mechanismof the cone penetration into a cohesive terrain, considering it as a limit equilibrium problem.In a somewhat different vein, Yong et al. [34] studied the mechanism of cone penetrationas a problem involving the mutual relationship between soil and wheel.

In general, where one has many passes of a piece of construction machinery in the samerut one gets remolding of a cohesive soil, with a consequent reduction of its shear strength.As a result the trafficability of the terrain will reduce.

The Japanese Association for Road Engineering’s criteria for terrain trafficability arebased on the cone index value that is necessary to allow a certain number of passes in thesame rut. Table 1.2 gives a lower limit cone index value to permit the trafficability of variouscategories of construction machinery. In contrast, Cohron [35] has proposed a trafficabilityprediction method based on a comparison between the rating cone index RCI and the vehiclecone index VCI [36] that is necessary for one pass of a piece of construction machinery.The VCI value may be calculated from the mobility index MI [37] for a particular type

Introduction 19

Table 1.2. Cone index necessary for trafficability of construction machinery.

Construction machine and situation Cone index CI (kN/m2)

Bulldozer running on very weak terrain ≥196Bulldozer running on weak terrain ≥294Middle size bulldozer ≥490Large size bulldozer ≥686Scrape-dozer running on weak terrain ≥392Scrape dozer ≥588Towed scraper ≥686Motor scraper ≥906Dump truck ≥1170

of construction machinery. The remolding index RI, which is the ratio of the rating coneindex RCI of the soil after it has been subjected to the remolding produced by 50 vehicularpasses and the cone index CI before remolding, is given by the following equation:

RI = RCI

CI(1.39)

In general, for straight forward running motion across a flat terrain, if:

RCI >VCI then vehicles can pass more than 50 times in the same rutRCI ≥VCI × 0.75 then vehicles can pass 1 ∼ 2 times in the same rutRCI ≤VCI × 0.75 then vehicles cannot pass.

1.3 MECHANICS OF SNOW COVERED TERRAIN

In this section we will present details of an in-situ experimental method that may be used toobtain snow characteristics and related snow physical properties. The method may also beused to obtain the snow’s essential compression and shear deformation properties. Theseproperties are necessary for calculation of the bearing capacity of a snow covered terrainor alternately of the thrust or drag of a tracked over-snow vehicle or wheeled snow removerduring driving or braking action on a snow covered terrain. The covered terrain can consistof loosely accumulated newly fallen fresh snow or older sintered snow. Sintered snow is asnow which has been hardened by metamorphic action due to temperature or aging.

1.3.1 Physical properties of snow

Snow is composed of three phases – ice (solid), pore water (fluid) and void air (air). A typicalcomposition is illustrated in Figure 1.16. Usually, snow can be classified into two types.The first type is wet snow at around 0 ◦C when there is liquid pore water present. The secondtype is dry snow which exists at under freezing temperature. In this case there is no liquidpore water present. After fresh snow has fallen on the ground, the density of the snow willincrease and become harder with time. This is due to the sublimation of the snow crystalparticles and to bonds developing between the snow particles. This collective phenomenon

20 Terramechanics

Figure 1.16. Idealised three-phase snow diagram.

is called snow metamorphism. During such a process a newly fallen snow of density of0.05 ∼ 0.10 g/cm3 can transform to a wet steady snow of density of 0.20 ∼ 0.40 g/cm3 andthence to a granular snow of density 0.30 ∼ 0.50 g/cm3.

Wet density ρw is defined as:

ρw = M

V(g/cm3) (1.40)

where M is the mass of the sample and V is the volume of the snow specimen as sampledby an angular or cylindrical snow sampler and spatula.

Dry density ρi is defined as:

ρi = Mi

Vρw

(100 − w

100

)g/cm3 (1.41)

where w is the total weight water content expressed as a percentage. The water content itselfis measured by use of a joint-type calorimeter which utilizes the latent heat of thawing ice.The final value may be calculated using the following equation:

w = M − Mi

M× 100 = Mw

M× 100 (%) (1.42)

where M is the mass of the wet snow, Mi is the mass of the dry snow and Mw is the massof the pore water.

Figure 1.17 shows the principle of operation of the widely used Yoshida joint-typecalorimeter [38]. Firstly, hot water of mass MA (g) and temperature T1 (◦C) is introducedinto a heat insulating vessel A where it is kept warm. Next, a wet snow sample of 0 ◦C andmass of MB (g) is taken from the snow covered area and put into another heat insulatingvessel B. After this, the two vessels are connected and shaken to thoroughly mix the hotwater and the snow sample. The temperature of the resulting mixture will become T2 (◦C). Ifthe water equivalents of the vessel A and B are mA (g) and mB (g) respectively, the followingidentity can be established:

(T1 − T2)(MA + mA) = T2 (MB + mB) + JMi (cal) (1.43)

where J is the latent heat of thawing ice (= 79.6 cal/g), and Mi (g) is the mass of ice in thewet snow sample MB. Rearranging this equation we get:

Mi = 1

J{(T1 − T2)(MA + mA) − T2 (M + mB)} (1.44)

Introduction 21

Figure 1.17. Measurement of water content of snow by use of a joint type calorimeter.

If one now substitutes Mi, calculated from the above equation, into Eq. (1.42), the watercontent w as a weight percentage can be obtained.

The degree of saturation Sr of a wet snow sample can be obtained from the ratio of volumeof pore water Vw to the total volume Vv of the pore water and the pore air. The relation isas follows:

Sr = Vw

Vv× 100 (1.45)

The hardness of a snow covered terrain can be obtained by measuring the penetration depthof a cone or circular plate when a weight is dropped onto it from a pre-determined height.The push-type hardness measurement apparatus [39] can also measure the hardness of asnow covered terrain by measuring the diameter of the opening made by a cone of 20 cmdiameter, height of 10 cm and apex angle of π/2 rad as it is spring-shocked by the apparatus.

A Canadian hardness meter, that is widely used in North America, measures the failureload of a snow covered terrain by use of a piece of equipment comprised of a circular plateattached to a cylinder that contains an in-built spring system. The plate is pushed into thesnow and penetrates it.

A further type of system is the Rammsonde – which is a kind of cone penetrometer. Inthis method, the depth of penetration of a cone of π/3 rad apex angle is measured when amass of 1 or 3 kg is dropped onto it from a prescribed height.

For the purpose of judging the trafficability of a snow remover or over-snow vehicle,Kinoshita’s hardness meter [40] is widely used. This apparatus is shown in Figure 1.18.A mass m is dropped n times from a height h onto a circular plate and then the amount ofsinkage d of the plate is measured. The plate has an area S and a mass M . The hardness ofthe snow covered ground H can then be computed by use of the following expression:

H = 1

S

{m

(1 + nh

d

)+ M

}(1.46)

This equation has been developed on the assumption that the total potential energies ofthe weight and the circular plate equate to the mechanical work done by the plate as itmoves the snow against the penetration resistance. The hardness H can be viewed as being

22 Terramechanics

Figure 1.18. Kinoshita’s hardness test apparatus.

the average penetration pressure under the circular plate. For naturally deposited snowand compacted snow, Kinoshita [41] concluded that the hardness of dry snow H (kPa) isproportional to the 4th power of the dry density ρ (g/cm3). That is:

H = cρ4 (1.47)

where the value of c is 100 for a naturally deposited snow and 400 ∼ 600 for a compactedsnow.

Yoshida’s experiment [42] explains why the hardness of wet snow decreases with anincrease in water content as a percentage of total weight.

1.3.2 Compressive stress and deformation characteristics

Under the passage of snow removers or tracked over-snow vehicles, snow will becomecompressed. It will become even more compressed over time as the snow particles becomebonded together by sintering. The relationship between the density of wet snow and the watercontent is very important in determining the engineering properties of compressed snow.

Figure 1.19 shows a series of results obtained where compaction of wet snow is obtainedthrough the application of a constant compaction energy. Compaction in this case is obtainedthrough use of special apparatus developed in conformance to JIS A 1210. From a studyof this Figure, it can be concluded that the wet density ρw (g/cm3) increases linearly withan increase in the total weight based, water content w (%) and can be expressed by theequation:

ρw = 9.98 × 10−3w + 0.54 (1.48)

In contrast, the dry density ρi, can be seen in Figure 1.20, to not change with a change of,total weight-based water content w and it is evident that ρi remains at a constant value ofρi = 0.57 ± 0.05 g/cm3.

This encountered phenomenon is completely different from that found in relation to soilmaterials which have a maximum dry density at an optimum water content.

Introduction 23

Figure 1.19. Relationship between wet density of wet snow compacted with constant energy andwater content [43].

Figure 1.20. Relationship between wet density of wet snow compacted with constant energy andwater content [43].

Next, let us consider the results that are obtained when a rectangular plate loading test [44]is performed on a snow covered terrain.

As shown in Figure 1.21, the main failure pattern resulting from a rectangular plateloading test on a dry fresh snow is that of the development of a compressed zone bcfesurrounded by two vertical slip lines abc and def . As shown in Figure 1.22, the generalrelationship established between penetration pressure p and penetration depth X for sixdifferent kinds of dry fresh snow – identified as A0, B0, C0, D0, E0 and F0 having variousinitial densities at minus 13 ◦C – shows an elastic behaviour at the initial stage. After that,a plastic compression zone of snow develops under the rectangular plate and considerablemicro-structural failures occur among the snow particles. These failures occur successivelyin front of the plastic compression zone. Subsequent to these two initial stages, we then geta phenomenon associated with plastic failure i.e. a saw-toothed hump. When the plastic

24 Terramechanics

Figure 1.21. Compression and deformation process of fallen snow for plate loading.

Figure 1.22. Relationship between penetration resistance, depth of penetration and compressivestrain.

compression zone reaches the bottom of the snow box, the penetration pressure increasesrapidly with increasing penetration depth.

In this case, the snow sample is assumed to be failed during the plate loading test sincethe strain controlled deformation rate of 3.7 mm/s is large enough as compared to a criticalrate proposed by Kinoshita [45]. These phenomena have been observed in the plate loadingtest of Yoshida et al. [46] and have been observed in triaxial compression tests carried outat comparatively large deformation rate – cf. Kawada et al. [47].

In the initial elastic phase of the process, the relationship between the elastic modulusE (kPa) and the initial density ρ0 (g/cm3) of snow is given by the following relation:

E = 2642ρ2.8260 (1.49)

Introduction 25

Table 1.3. Constant values from plate loading test for various snow samples.

Sample ρd E K Xth εth ρth Xf εf ρf

(g/cm3) (kPa) (cm) (%) (g/cm3) (cm) (%) (g/cm3)

A0 0.418 ± 0.017 153.4 3.800 5.72 20.9 0.512 ± 0.005 10.46 38.2 0.575B0 0.352 ± 0.022 213.7 3.000 6.86 25.0 0.469 ± 0.007 9.70 35.4 0.515C0 0.184 ± 0.017 43.8 5.353 4.32 15.8 0.229 ± 0.003 10.92 39.9 0.253D0 0.391 ± 0.014 104.1 4.425 5.08 18.5 0.467 ± 0.003 13.34 48.7 0.487E0 0.199 ± 0.021 16.4 5.750 5.33 19.5 0.262 ± 0.002 14.22 51.9 0.333F0 0.325 ± 0.010 76.7 4.564 4.95 18.1 0.390 ± 0.002 14.22 51.9 0.333D1 0.406 ± 0.001 2529.0 3.154 6.60 24.1 0.524 ± 0.003 – – –D3 0.397 ± 0.001 2058.0 2.708 – – 0.491 ± 0.003 – – –D7 0.398 ± 0.002 5055.0 2.500 – – 0.512 ± 0.007 – – –D14 0.398 ± 0.011 8505.0 2.200 – – 0.522 ± 0.004 – – –

Because the thickness of the plastic compression zone is observed to be proportional to thepenetration depth X , the coefficient of proportionality K can be regarded as a coefficientof propagation of plastic compression. Under these conditions the relationship between Kand the initial density ρ0 (g/cm3) of snow can be expressed as follows:

K = 2.850ρ−3.9330 (1.50)

When the plastic compression zone under the rectangular plate reaches the bottom of asnow covered ground of depth H , the penetration pressure p increases rapidly, takes a peakvalue at a penetration depth Xth and then decreases temporarily. The initial density of snowρ0 changes to a threshold density, as ρth proposed by Yong et al. [48] within the plasticcompression zone and then it increases when the penetration depth X exceeds Xth.

The initial density finally develops to the density ρc of the polycrystallised ice whenX = Xth. In this process, another compressive shear failure may occur due to the develop-ment of lateral plastic flow. If this happens, then the penetration pressure p will increasedramatically. Table 1.3 lists a number of constants for a variety of snow samples. The Tablegives values for initial density ρ0, elastic modulus E, the coefficient of propagation of plas-tic compression K , threshold penetration depth Xth, threshold strain εth = Xth/H of the snowcovered ground as well as threshold density ρth, final penetration depth of the rectangularplate Xf , final strain ε = Xf /H of the snow covered terrain, and the final density ρf for anumber of snow samples. Also given in the Table are the results of a series of rectangularplate loading tests for a number of sintered snow samples: D1, D3, D7 and D14 (wherethe subscripted numbers denote the age of sintering) for a snow sample D conditionedunder a temperature of minus 13 ◦C. As shown in the Table, penetration resistance undera constant penetration depth of the plate increases with increasing age of sintering. Thefundamental failure pattern of the sintered snow can be explained in a similar way to thatalready mentioned in Figure 1.21 in that, it shows a remarkable elastic behaviour at theinitial stage. After that, the failure pattern shows a complex behaviour as the snow changesits rheology and goes from an elasto-plastic to a rigid-plastic material as the penetrationdepth reaches some value.

26 Terramechanics

1.3.3 Shear stress and deformation characteristics

To calculate the thrust or drag of a snow remover or tracked over-snow vehicle during drivingor braking action, it is necessary to fully understand the shear deformation characteristicsof snow covered terrains. Yong et al. [49] in 1978 performed a number of shear tests fora variety of snow materials. In their tests they used a shear box and established that themechanism of shear in snow can be explained by consideration of the mutual adhesion andthe mutual frictional forces that exist between the individual snow particles.

They also hypothesised that the Coulomb frictional relationship that connects normalstress and shear resistance could be used for artificially crushed snow particles but then therule would not apply to sintered snows because the snow particles are bonded tightly dueto sintering action.

Now, let us consider a test method for prediction of snow shear resistance based on theidea of use of a vane cone. Photo 1.1 shows a vane cone test apparatus as set up in a coldroom of temperature of minus 13 ◦C.The test procedure involves making a sample by fillingup a snow box. The vane or a vane cone as shown in Photo 1.2 is then pushed into the snowsample at constant penetration speed.

After measuring the relationship between the penetration force F and the penetrationdepth, another relationship between the torque T and the rotational deformation X can bedetermined. These factors can be measured as the vane or vane cone rotates and shears thesnow sample. The test conditions are set at a peripheral speed of 1.89 mm/s and measure-ments are taken at a variety of depths.

Photo 1.1. Vane cone test apparatus.

Introduction 27

The merit of the vane cone test is that the relationship between the shear resistance τ andthe shear deformation X can be measured directly under a constant normal stress σ.

In general, the shear resistance τ shows the shear deformation characteristic of a rigidplastic material in which the shear resistance goes immediately to a maximum value τmax

without any shear deformation and then decreases rapidly with increasing shear deformationX . The material follows the relation:

τ = τmax exp(−αX ) (1.51)

where α is a constant value which is determined by the snow sample and the shear rate etc.If one assumes that the distribution of shear resistance τ acting on the vertical peripheralshear surface and the horizontal bottom shear surface of the vane cone of height of H andof radius r becomes uniform, then the normal stress σ and the shear resistance τ can becalculated from the penetration force F and the maximum torque Tmax. This latter factoris measured in correspondence to the maximum shear resistance τmax at X = 0 [50]. Thecalculated results can then be developed by use of the following equations.

σ = F

πr2(1.52)

τmax = Tmax

2π(r2H + r3/3)(1.53)

As an example, Figure 1.23 shows the relationship between normal stress σ and shearresistance τ for a snow sample C [51].

Table 1.4 shows the relationships that exist between σ and τ for 6 kinds of fresh snowand for some samples of sintered snow. This data confirms that the Mohr-Coulomb failurecriterion operates for the whole range of initial density, threshold density and final densityof the fresh snow and for sintered snow samples. These results arise because the soil samplesare tested in a state of normal compression as a consequence of the collapse of the structureof snow particles during the cone penetration.

Next let us consider the shear deformation characteristics of wet snow. Photo 1.3 showsa portable vane cone test apparatus. The shear deformation characteristics for a wet snowhaving the initial density of ρ0 = 0.228 g/cm3 were measured in the field using this appa-ratus. The experimental study showed the existence of an initial elasto-plastic behaviour

Photo 1.2. Vane and vane cone.

28 Terramechanics

Figure 1.23. Relationship between shear stress and normal stress for the vane cone test [51].

Table 1.4. Relationship between shear stress and normal stress.

Sample τ = f (σ) kPa

A0 τ = 2.5 + 0.103 σB0 τ = 3.4 + 0.277 σC0 τ = 1.0 + 0.177 σD0 τ = 4.3 + 0.418 σE0 τ = 2.0 + 0.390 σF0 τ = 2.1 + 0.548 σD1 τ = 6.1 + 0.214 σD3 τ = 8.8 + 0.245 σD7 τ = 17.3 + 0.348 σD14 τ = 13.6 + 0.222 σ

Photo 1.3. Portable vane cone apparatus.

Introduction 29

Photo 1.4. Slip line of snow vane cone test.

phase, in which the shear resistance took a maximum value at some shear deformation, andafter that it decreased suddenly [52]. In this case, the slip line in the snow that developedduring the vane cone shear test developed along a logarithmic spiral line. This spiral failureline is clearly visible in Photo 1.4. This shows clearly that snow materials are a kind of φ

material.The shear stress and deformation characteristics for the same wet snow were measured

using the same portable shear test apparatus as shown in Figure 1.24 but to which wasattached a ring of inner diameter of 0.5 cm and outer diameter of 3.5 cm and upon whichwas mounted a set of six vanes of height 1.0 cm.

Using this apparatus, the relationship between the shear resistance τ and the shear defor-mation y for the wet snow terrain – of threshold density ρth = 0.275 g/cm3 and the criticaldensity ρC = 0.600 g/cm3 – could be experimentally explored. In this case, the study showedthat the material behaves, phenomenologically, as an elasto-plastic material. Characteristicresults for the tests are shown in Figure 1.25. In this case, the relationship between normalstress and the shear resistance also satisfies Coulomb’s failure criterion. The angle of inter-nal friction measured in this ring shear test is generally larger than that derived from thevane cone shear test for remolded snow samples.

1.4 SUMMARY

In this chapter we have introduced the idea of terramechanics as a study discipline andhave located it at the interface between the two domains of machinery design/operation andground behaviour.

We have also reviewed some traditional matters of soil mechanics as a precursor to apply-ing them to machinery-terrain interaction problems. Also, we have analysed the mechanical

30 Terramechanics

Figure 1.24. Portable ring shear test apparatus.

Figure 1.25. Relationship between shear stress and shear deformation.

Introduction 31

properties of snow in some detail – since snow is a particular kind of terrain that is veryimportant in some countries and in some latitudes.

Having studied this chapter, the reader should be able to describe the mechanical pro-cesses typically employed to, objectively and quantitatively, characterise specific terrains.The reader should also be able to appreciate and comment upon the degree to which thevarious terrain metrics can be used as inputs to engineering prediction models. These pre-diction models can be used for on-surface vehicle running behaviour prediction and/or forterrain trafficability studies.

REFERENCES

1. The Japanese Geotechnical Society (1983). Soil Testing Methods. (pp. 82–116). (In Japanese).2. Yamaguchi, H. (1984). Soil Mechanics. (pp. 9–13). Gihoudou Press. (In Japanese).3. Akai, K. (1986). Soil Mechanics. (pp. 3–26). Asakura Press. (In Japanese).4. Oda, M., Enomoto, H. & Suzuki, T. (1971). Study on the Effect of Shape and Composition of

Sandy Soil Particles on Soil Engineering Properties. Tsuchi-to-Kiso, Vol. 19, No. 2, 5–12. (InJapanese).

5. Toriumi, I. (1976). Geotechnical Engineering. (pp. 26–51). Morikita Press. (In Japanese).6. Akai, K. (1964). Bearing Capacity and Depression of Soil. (pp. 5–34). Sankaido Press. (In

Japanese).7. Akai, K. (1976). Advanced Soil Mechanics. (pp. 201–234). Morikita Press. (In Japanese).8. Terzaghi, K. & Peck, R.B. (1960). Soil Mechanics in Engineering Practice. (pp. 413–443). John

Wiley & Sons and Charles E. Tuttle.9. The Japanese Geotechnical Society (1982). Handbook of Soil Engineering. (pp. 484–486). (In

Japanese).10. Bekker, M.G. (1960). Off-the-Road Engineering. (pp. 25–40). The University of Michigan Press.11. Sugiyama, N. (1982). Several Problems between Construction Machinery and Soil.

(pp. 105–113). Kashima Press. (In Japanese).12. Taylor, D.E. (1984). Fundamentals of Soil Mechanics. John Wiley & Sons.13. Wong, J.Y. (1989). Terramechanics and Off-road Vehicles. (pp. 29–71). Elsevier.14. Reece, A.R. (1964). Principles of Soil-Vehicle Mechanics. Proc. Instn Mech. Engrs. 180, Part

2A(2), 45–67.15. The Japanese Geotechnical Society (1983). Soil Testing Methods. (pp. 433–469). (In Japanese).16. Leonards, G.A. (1962). Foundation Engineering. (pp. 176–226). McGraw-Hill and Kogakusha.17. Kondo, H., Noda, Y. & Sugiyama, N. (1986). Trial Production of Dynamic Direct Shear.

Terramechanics, 6, 98–104. (In Japanese).18. Bjerrum, L. (1973). Problems of Soil Mechanics and Construction on Soft Clays and Structurally

Unstable Soil (Collapsible, Expansive and Others). Proc. of the 8th Int. Conf. on Soil Mechanicsand Foundation Engineering, Vol. 3, 111–159.

19. Richardson, A.M., Brand, E.W. & Memon, A. (1975). In-situ Determination of Anisotropy ofa Soft Clay. Proc. of the Conf., In-situ Measurement of Soil Properties. ASCE, Vol. 1, 336–349.

20. Shibata, T. (1967). Study on Vane Shear Strength of Clay. Proc. JSCE, No. 138, 39–48. (InJapanese).

21. Muro, T. & Kawahara, S. (1986). Interaction Problem between Rigid Track and Super-WeakMarine Sediment. Proc. JSCE, No. 376/III-6. (In Japanese).

22. Yonezu, H. (1980). Experimental Consideration on Laboratory Vane Shear Test. Tsuchi-to-kiso,Vol. 28, No. 4, 39–46. (In Japanese).

23. Umeda, T., Ooshita, K., Suwa, S. & Ikemori, K. (1980). Investigation of Soft Ground using Vaneof Variable Rotational Speed. Proc. of the Symposium on Vane Test. (pp. 91–98). The JapaneseGeotechnical Society. (In Japanese).

32 Terramechanics

24. Söhne, W. (1961). Wechselbeziehungen zwischen fahrzeuglaufwerk und boden beim fahren aufunbefestigter fahrbahn. Grundlagen der Landtechnik, Heft 13, 21–34.

25. Tanaka, T. (1965). Study on the Evaluation of Capability of Tractor on Paddy Field. Journal ofAgricultural Machinery, 27, 3, 150–154. (In Japanese).

26. Bekker, M.G. (1969). Introduction to Terrain-Vehicle Systems. (pp. 3–37). The University ofMichigan Press.

27. Golob, T.B. (1981). Development of a Terrain Strength Measuring System. Journal ofTerramechanics, 18, 2, 109–118.

28. Wong, J.Y. (1980). Data Processing Methodology in the Characterization of the MechanicalProperties of Terrain. Journal of Terramechanics, 17, 1, 13–41.

29. Desai, C.S. & Siriwardane, J. (1984). Constitutive Laws for Engineering Materials with Emphasison Geologic Materials. Prentice-Hall.

30. Mogami, T. (1969). Soil Mechanics. (pp. 492–506). Gihoudo Press. (In Japanese).31. Skempton, A.W. & Bishop, A.W. (1954). Soils (pp. 417–482). Chapter 10 of Building Materials.

Amsterdam: North Holland.32. Muromachi, T. (1971). Experimental Study on the Application of Static Cone Penetrometer to

the Investigation of Soft Ground. Report on RailRoad Engineering, No. 757. (In Japanese).33. Hata, S. (1964). On the Relation between Cone Indices and Cutting Resistances of Cohesive

Soils. Proc. JSCE, No. 110, 1–5. (In Japanese).34. Yong, R.N., Chen, C.K. & Sylvestre-Williams, R. (1972). Study of the Mechanism of Cone

Indentation and its Relation to Soil-Wheel Interaction. Journal of Terramechanics, Vol. 9, No. 1,19–36.

35. Cohron, G.T. (1975). A New Trafficability Prediction System. Proc. 5th Int. Conf., Vol. 2, ISTVS,(pp. 401–425).

36. Sugiyama, N. (1982). Several Problems between Construction Machinery and Soil. (pp. 90–96).Kashima Press, (In Japanese).

37. Tire Society (1969). Considerations on Tire for Construction Machinery. Komatsu TechnicalReport, Vol. 15, No. 2, 1–54.

38. Yoshida, Z. (1959). Thermodynamometer Measuring Water Content of Wet Snow. Institute ofLow Temperature Science, Hokkaido University, A-18, (pp. 19–28). (In Japanese).

39. Japanese Construction Machinery Association (1977). New Handbook of Snow PreventionEngineering. (pp. 11–58). Morikita Press. (In Japanese).

40. Kinoshita, S. (1960). Hardness of Snow Covered Terrain. Institute of Low Temperature Science,Hokkaido University, Physics, Vol. 19, (pp. 119–134). (In Japanese).

41. Kinoshita, S., Akitaya, E. & I. Tanuma (1970). Investigation of Snow and Ice on Road II.Institute of Low Temperature Science, Hokkaido University, Physics, Vol. 28, (pp. 311–323).(In Japanese).

42. Yoshida, Z. (1974). Snow and Construction Works (pp. 25–41). Journal of the Japanese Societyof Snow and Ice. (In Japanese).

43. Muro, T. (1978). Compaction Properties of Wet Snow. Journal of the Japanese Society of Snowand Ice, Vol. 40, No. 3, (pp. 110–116). (In Japanese).

44. Muro, T. & Yong, R.N. (1980). Rectangular Plate Loading Test on Snow – Mobility of TrackedOversnow Vehicle. Journal of the Japanese Society of Snow and Ice, Vol. 42, No. 1, 17–24.(In Japanese).

45. Kinoshita, S. (1967). Compression of Snow at Constant Speed. Proc. of 1st Int. Conf. on Physicsof Snow and Ice, Vol. 1, No. 1, (pp. 911–927). The Institute of Low Temperature Science,Hokkaido University, Sapporo, Japan.

46. Yoshida, Z., Oura, H., Kuroiwa, D., Hujioka, T., Kojima, K., Aoi, S. & Kinoshita, S. (1956). Phys-ical Studies on Deposited Snow, II. Mechanical properties (1), Vol. 9, (pp. 1–81). Contributionsfrom the Institute of Low Temperature Science, Hokkaido University.

47. Kawada, K. & Hujioka, T. (1972). Snow Failure observed from Tri-Axial Compression Test.Physics, Vol. 30, (pp. 53–64), Institute of Low Temperature Science, Hokkaido University. (InJapanese).

48. Yong, R.N. & Fukue, M. (1977). Performance of Snow in Confined Compression. Journal ofTerramechanics, Vol. 14, No. 2, 59–82.

Introduction 33

49. Yong, R.N. & Fukue, M. (1978). Snow Mechanics – Machine Snow Interaction. Proc. of the 2ndInt. Symp. on Snow Removal and Ice Control Research, (pp. 9–13).

50. Muro, T. & Yong, R.N. (1980). Vane Cone Test of Snow-Mobility of Tracked Oversnow Vehicle.Journal of the Japanese Society of Snow and Ice, Vol. 42, No. 1, 25–32. (In Japanese).

51. Yong, R.N. & Muro, T. (1981). Plate Loading and Vane-Cone Measurements for Fresh andSintered Snow. Proc. of the 7th Int. Conf. of ISTVS, Calgary, Canada, Vol. 3, (pp. 1093–1118).

52. Muro, T. (1984). Shallow Snow Performance of Tracked Vehicle. Soils And Foundations, Vol. 24,No. 1, 63–76.

EXERCISES

(1) The water content of a soil sampled in the field can be calculated from the differencebetween the weight of a wet and an oven dried soil sample. Suppose then, that theinitial weight of a moist soil sample plus container was measured to be 58.2 g. Also,suppose that the total weight of the sample plus container – after oven drying for aperiod of 24 hours – was 50.6 grams. If the self weight of the container was 35.3 g,calculate the water content of the soil sample.

(2) Suppose that the grain size distribution curve for a particular soil sample is as shown inFigure 1.2, calculate the coefficient of uniformity Uc and the coefficient of curvatureU ′

c of the soil.(3) Imagine that a plate loading test has been carried out to simulate a tracked machine’s

operation over a sandy terrain. The dimensions of the rectangular plate were: length50 cm, width 20 cm and depth 5 cm. Assume also that results for the terrain systemconstants for Bekker’s equation (1.19) which develops a relationship between the con-tact pressure p and the amount of sinkage s0, have been obtained as kc = 18.2 N/cmn+1,kφ = 5.2 N/cmn+2 and n = 0.80 respectively. Using this data, calculate the staticamount of sinkage s0 of a bulldozer of track length 1.5 m, track width 20 cm, andmean contact pressure 34.3 kPa.

(4) Suppose that a vane of diameter 5 cm and height of 10 cm is forced into a clayey terrain.After penetration the vane was rotated and a maximum torque value Mmax = 520 N/cmwas obtained. Calculate the cohesion c of the clay, neglecting any adhesion betweenthe rod of the vane and the clay.

(5) Given the following relationship between shear resistance τ and amount of slippagej, calculate the relationship between the modulus of deformation E and the constantvalue a.

τ = τmax {1 − exp(−aj)}(6) Calculate the maximum amount of slip sinkage smax when a tracked model machine

of length 100 cm and width 20 cm loaded with a weight of 4410 N is pulled overa sandy terrain. The terrain-track system constants as given in Eq. (1.30) arec0 = 0.00476 cm2c1−c2+1/Nc1 , c1 = 2.07, and c2 = 1.07.

(7) To measure the weight percentage water content of a newly fallen wet snow, a jointtype calorimeter as shown in Figure 1.17 was used and 2500 g of water at 85 ◦C wasprepared and put into vessel A. Next, 300 g wet snow of 0 ◦C was sampled and put intovessel B. After joining both the vessels, the container was shaken until the temperatureof the hot water assumed a constant value. If the temperature of the hot water decreased

34 Terramechanics

to 55 ◦C due to melting of the snow sample, calculate the total weight based percentagewater content of the wet snow. Assume a water equivalent of 400 g for vessel A and300 g for vessel B.

(8) To measure the hardness of a snow covered terrain, Kinoshita’s hardness testingmachine is widely used. The mass of the falling weight is m, and the base area ofthe disc is S and its mass is M . Assuming that the total energy, when the falling weightfalls n times from a constant height h equals the energy of penetration of the disc intothe snow covered terrain, develop an equation to calculate the hardness H of the snowcovered terrain.

(9) Assume that the total-weight based water content of a wet snow sample is measuredas 20%. When this wet snow sample is compacted mechanically by a falling rammerat a constant compaction energy based on the criterion of JIS A 1210, calculate thecompacted wet density ρw and the compacted dry density ρd of the snow sample.

(10) Suppose that a vane cone of radius of vane r = 3 cm and height H = 12 cm ispenetrated fully into a snow covered terrain and that a maximum torque value ofTmax = 2.30 kN/cm is measured under a penetration force of F = 332 N. Calculate thenormal stress σ and the maximum shear resistance τmax acting on the snow coveredground.

Chapter 2

Rigid Wheel Systems

For the compaction of subgrade materials, asphalt pavement materials and roller compactedconcrete, several forms of compaction machines equipped with single steel drum rollersand tandem rollers are in everyday use in civil engineering – for example the macadam andsteam rollers of old. The ‘wheels’ on these machines do not deform to any degree whenthey accept axle load, and as a consequence are referred to as ‘rigid wheels’.

Al-Hussaini et al. [1] have analysed the distribution of contact pressures that will existunder a rigid wheel whilst it is running on a hard terrain. They analysed the distributionof normal stress and shear resistance using a procedure based on Boussinesq’s theory. InBoussinesq’s theory, the ground is assumed to be a semi-infinite elastic material. Theseresearchers concluded that a network of stress lines corresponding to those of an equivalentnormal and shear stress developed within the ground under a rigid wheel. In other researchwork, Ito et al. [2] measured the distribution of normal stress acting around a rigid wheelduring driving action on a hard medium by use of a photo-elastic epoxy resin. They showedthat the distribution of contact pressure could be divided into two parts; one is a compressionzone which develops as a result of the thrust of wheel and the other is a bulldozing zonewhich produces increased land locomotion resistance with increasing vehicle sinkage.

Relative to the distribution of contact pressure of a rigid wheel running on a weakterrain, Wong et al. [3] showed that the application point of the maximum normal stressthat develops on a sandy soil terrain could be expressed as a linear function of the slip ratio.They also showed that the distribution of shear resistance could be calculated from the slipratio and that the normal stress distribution could be assumed to be symmetrical around themaximum-value application point .

Yong et al. [4–5] further analysed the distribution of contact pressure under a rolling rigidwheel by use of a theory of visco-plasticity that includes the strain rate behaviour of the soil.They also developed a method for predicting the continuous behaviour of a rigid wheel byuse of the principle of energy conservation and equilibrium. They also attempted to analysethe stress response under a rigid running wheel using the finite element method (FEM).However, despite this research there are many unresolved problems which still need to beaddressed in the future. These arise because it is still difficult to estimate the stress–strainrelationship under a rigid wheel from the results of the plate loading test.

The trafficability of a rigid wheel running on a shallow snow covered terrain may becalculated from the total of the compaction energy – which is necessary to collapse theunremolded fresh snow under the roller – and the slippage energy which is required togive thrust to the wheel supplemented by the local melting of snow due to an excessiveslip of wheel. Using this broad method, Harrison [6] developed a relationship between theeffective tractive effort, amount of sinkage and the compaction resistance of a rigid wheelrunning over snow materials.

36 Terramechanics

In the following sections of this Chapter, we will consider the various modes of behaviourof a rigid wheel running on weak ground during driving or braking action. We will theninvestigate the fundamental mechanics of land locomotion in relation to driving or brakingforce, thrust or drag, land locomotion resistance, effective driving or braking force, amountof sinkage and distribution of contact pressure.

2.1 AT REST

2.1.1 Bearing capacity of weak terrain

Where it is required to avoid an excessive amount of sinkage of a rigid wheel, such as whenthe wheel passes over a very weak terrain or a snowy ground, one needs to be able to predictthe ground bearing capacity. The ground bearing capacity of a rigid wheel at rest Qw can becalculated by use of the following formula for bearing capacity [7]. The formula assumesthat the contact shape of the roller can be modelled as a rectangular plate of wheel width Band contact length L.

Qw = BL

{(1 + 0.3

B

L

)cNc +

(0.5 − 0.1

B

L

)BγNγ + γDf Nq

}(2.1)

In this expression c and γ are the cohesion and unit weight of soil and γDf is the surchargecorresponding to the amount of sinkage of wheel Df . The factors Nc, Nγ and Nq are theTerzaghi coefficients of bearing capacity [8]. The factors may be determined from thecohesion and the angle of internal friction of the soil.

Relative to the stress conditions that prevail under a static rigid wheel, Terzaghi [9]predicted that the normal stress σZ in the vertical direction at a depth z under the cornerof rectangular plate B × L to which is applied a uniformly distributed load q could becalculated as:

σZ = q · Iσ

Iσ = 1

4π

{2mn(m2 + n2 + 1)

12

m2 + n2 + m2n2 + 1· m2 + n2 + 2

m2 + n2 + 1+ tan−1 2mn(m2 + n2 + 1)

12

m2 + n2 − m2n2 + 1

}(2.2)

where m = B/z and n = L/z are non-dimensional factors and Iσ is referred to as the factor ofinfluence. The normal stress σ in the vertical direction at an arbitrary point in the interiorof a rectangular plate may be calculated by a method that involves the sub-division ofrectangles.

2.1.2 Contact pressure distribution and amount of sinkage

When a rigid wheel has an axle load W applied to it while it is standing at rest on a weakterrain, a reaction force develops that is a product of a symmetrical distribution of normalstress σ acting on the peripheral surface of the wheel in the normal direction and a shearresistance τ acting in the tangential direction of the wheel. Physically, the sign of the shearresistance τ should be reversed for the left and right hand sides of the wheels, such thatthe torque Q i.e. the integration of the shear resistance for the whole range becomes zero.A zero summation is required since no net torque on the wheel may be present.

Rigid Wheel Systems 37

Figure 2.1. Distribution of ground reaction under a rigid wheel at rest.

If one considers the resultant stress p that develops between the normal stress σ and theshear resistance τ at an arbitrary point X along the peripheral surface of the contact part ofwheel and terrain, the direction of application of the resultant stress p is must be inclined atan angle δ = tan−1(τ/σ) relative to the normal direction. The general situation is as shownin Figure 2.1. The values of τ and δ depend on the amount of static slippage j between therigid wheel and the terrain.

In this diagram, R is the radius of the rigid wheel, B is its width, 2θ0 is the central angle∠AOE, and θ is the central angle of an arbitrary point X on the contact part of the wheel.For the amount of sinkage s0 of the bottom-dead-center of the wheel, the amount of sinkagez at the arbitrary point X can be calculated from geometry to be:

z = R(cos θ − cos θ0) (2.3)

Substituting θ = 0 in the above equation, s0 can be determined as,

s0 = R(1 − cos θ0) (2.4)

When a rigid wheel penetrates into a ground under static load conditions, an interfacialshear resistance τ develops along the peripheral surface of the wheel to left and to the righthand sides of the center line. This interface tangential force develops by virtue of a relativemotion (i.e. a slippage) that develops at the interface between the soil and the non-yieldingpenetrating wheel.

In the stationary wheel case, the amount of slippage j0 can be assumed to develop as afunction of the center angle θ as:

j0 = R(θ0 − sin θ0)θ0 − θ

θ0(2.5)

38 Terramechanics

In general, the Mohr-Coulomb failure criterion for the maximum shear resistance τmax

to the normal stress can be used to analyse the interaction problem between a soil and arigid wheel. In this situation:

τmax = ca + σ tan φ (2.6)

Here, ca and φ are parameters representing the adhesion and the angle of friction betweenthe rigid wheel and the ground, respectively. The numerical value of these factors dependson the material that comprises the surface of the wheel, its surface roughness and the soilproperties.

Most commonly, the shear resistance τ that develops along a rigid wheel running on aweak sandy or clayey terrain is smaller than τmax and follows a relation as per Figure 2.2where:

τ = c′a + σ tan φ′ (2.7)

The parameter τ is the mobilized shear resistance which develops for a relative amount ofslippage j between the rigid wheel and the terrain. It can be expressed as the product of aslippage function f ( j) times τmax as follows:

τ = τmax f ( j) (2.8)

The slippage function f ( j) for a loose sandy soil and a weak clayey soil was given byJanosi-Hanamoto [10] as follows:

f ( j) = 1 − exp(−aj) (2.9)

In this expression the constant a is the value of the coefficient of deformation of the soildivided by the maximum shear strength τmax.

For a hard compacted sandy soil and for a hard terrain, Bekker [11] and Kacigin [12]have proposed another function.

Working from these relations, the shear resistance τ(θ) applied under the rigid wheel canbe calculated as a function of the central angle θ as follows:

τ(θ) = {(ca + σ(θ) tan φ)} {1 − exp(−aj0(θ))} (2.10)

Again, the vertical component q(θ) of the resultant stress p(θ) can be calculated as a functionof the central angle θ from the plate loading test results through the use of Eq. (1.19). Note

Figure 2.2. Relations between τmax, τmob.


that allowance must be made here for the size effect of the contact length b. The result isthe following expression:

p(θ) = q(θ)

cos(θ − δ)= k1zn1

cos(θ − δ)=(

kc1

b+ kφ1

)Rn1

(cos θ − cos θ0)n1

cos(θ − δ)(2.11)

Thence, the axle load W applied on the rigid wheel can be calculated by integration of thevertical component of the resultant stress p(θ) from θ = −θ0 to θ = +θ0 as follows:

W = BR∫ θ0

−θ0

p cos(θ − δ) cos θ dθ (2.12)

Here, the central angle θ0 i.e. the entry angle is required to be determined. First of all, thedistribution of normal stress σ(θ) = p(θ)cos δ and shear resistance τ(θ) = p(θ) sin δ have tobe calculated for a given angle θ0 and δ, and then the angle δ can be determined exactlyfrom δ(θ) = tan−1{τ(θ)/σ(θ)}.

These distribution of σ(θ) and τ(θ) can be iteratively calculated until the real distributionof σ(θ), τ(θ) and δ(θ) is determined. Thereafter, the entry angle θ0 can be determined by arepetition of calculation using the two division method until the axle load W is determined.

Then, the real entry angle θ0, the real amount of sinkage s0, and the distribution of normalstress σ and shear resistance τ can be determined.

As a further comment, it is noted that a horizontal force does not occur on a static rigidwheel, because the horizontal component of the resultant stress p is symmetrical on boththe left and right hand sides.

2.2 AT DRIVING STATE

2.2.1 Amount of slippage

Figure 2.3 shows the distribution of slip velocity and amount of slippage respectively, forthe contact part of a rigid wheel when it is in a driving state.

For a wheel of radius R, a moving speed V of the wheel in the direction of the terrainsurface, and an angular velocity ω = −(dθ/dt), the tangential slip velocity Vs at an arbitrary

Figure 2.3. Distribution of slip velocity Vs and amount of slippage jd during driving action (Rω > V ).

40 Terramechanics

point X of central angle θ on the peripheral surface in the contact part of wheel to the terraincan be expressed as:

Vs = Rω − V cos θ (2.13)

where the angle θ is the angle measured from OM to the radius vector OX. It is defined aspositive for a counter clockwise direction. As Rω > V for the driving state, the slip ratioid is defined as follows:

id = 1 − V

Rω(2.14)

By substituting the above slip ratio id into Eq. (2.13), the slip velocity Vs at a point X canbe expressed as:

Vs = Rω{1 − (1 − id) cos θ} (2.15)

After Vs takes a maximum value at the entry angle θ = θf , Vs decreases gradually to aminimum value Rω − V at the bottom-dead-center M and then increases slightly towardsthe point E at exit angle θ = −θr .

Integrating the slip velocity Vs from the beginning of contact t = 0 to an arbitrary timet = t, the amount of slippage jd can be calculated as follows:

jd = R∫ t

0ω {1 − (1 − id) cos θ} dt = R

∫ θf

θ

{1 − (1 − id) cos θ} dθ

= R{(

θf − θ)− (1 − id)

(sin θf − sin θ

)}(2.16)

The amount of slippage jd is positive for the whole range of the contact part AE. It increasesvery rapidly from zero at point A at entry angle θ = θf . It takes a maximum value at pointE at exit angle θ = −θr .

2.2.2 Soil deformation

Figure 2.4 shows the general soil deformation and the occurrence of slip lines when a rigidwheel is running on a sandy terrain during driving action. Typically, two slip lines emergefrom the point N on the contact interface between the rigid wheel and the terrain. Asdeveloped in Terzaghi’s theory of bearing capacity [13] the shape of the individual sliplines consists of a logarithmic spiral region and a straight line region.

When the slip line is in the region of the logarithmic spiral portion, the slip zone isreferred to as the transient state zone. Similarly, the slip zone surrounding the straight slipline is called the passive state zone. The straight slip line usually crosses the terrain surfaceat an angle π/4 − φ/2 for an angle of internal friction φ. As shown in Figure 2.4(a), thetwo slip lines s1 and s2 occur on both the left and right hand sides of point N. When thedirection of one slip line s1 is defined in the direction of the resultant velocity vector of Vand Rω, (which should occur at right angles to the moving radius IN for the instantaneouscenter N, the slip lines s1 and s2 cross at an angle ±(π/4 − φ/2) to the major principalstress line σ1 at the point N and at an angle ±(π/4 + φ/2) to the minor principal stressline σ3. The slipping soil mass NCA flows forward due to the down and forward resultant


Figure 2.4. Soil deformation under a rigid wheel during driving action.

velocity vector, whilst the another slipping soil mass NDE flows backward due to the downand backward resultant velocity vector.

Within both slipping soil masses, there are a lot of conjugate slip lines s1 and s2 whichintroduce the occurrence of plastic flow within the soil. In addition, it should be noticedthat the circumferential surface of the wheel is coincident directly with the slip line s1 atthe bottom-dead-center M.

The position of the point N and the direction of the slip lines depend on the slip ratioid which is calculated from the running speed V and the circumferential speed Rω of thewheel. In relation to this, Wong et al. [14] determined experimentally that the normal stressσ acting on a rigid wheel running on a sandy terrain takes a maximum value at point N.They also developed an experimentally based equation where the position φN of the point

42 Terramechanics

N divided by the entry angle φf could be expressed as a linear function of slip ratio id asfollows:

φN /φf = a + bid (2.17)

With reference to Figure 2.4(b), the angle η between the slip line s1 and the tangential lineat the point N can be calculated.

For the angle δ between the direction of resultant stress p and the normal line at the pointN and for the angle ξ between the slip line s1 and the resultant stress p:

tan η = V sin θ

Rω − V cos θ= (1 − id) sin θ

1 − (1 − id) cos θ

ξ = π

2− δ − η

∴ (1 − id) sin θ

1 − (1 − id) cos θ= cot(ξ + δ) (2.18)

As the angle ξ becomes the angle φ when the combination of the normal stress p cos ξ andthe shear resistance ±p sin ξ on the slip lines s1, s2 satisfies the failure condition of the soil,the angle θN of the point N can be derived theoretically using the above condition.

The resultant velocity vectors of all the soil particles on the contact part of the rigid wheelare always rotating around the instantaneous center I. The position of the center I is locatedbetween the axis O and the bottom-dead-center M for the driving state and the length OIcan be given as follows, using the symbols of Figure 2.4.

OI = R cos θ + R sin θ tan α = R cos θ + 1

ω(V − Rω cos θ)

= V

ω= R(1 − id) (2.19)

From the above equation, the length OI becomes the radius of wheel R when V = Rω

i.e. id = 0%, so the position of the instantaneous center I is coincident with the bottom-dead-center M. When the slip ratio id becomes 100%, OI = 0, then the position of I iscoincident with the axle point O. Therefore, the position of the center I depends on theslip ratio id , V and ω. This relationship between position of I and slip ratio id has beenvalidated experimentally by Wong [15]. From general mathematics theory, it can be seenthat the angle ß between the resultant velocity vector and the normal line at the point X asshown in Figure 2.4(a) takes a minimum value when the point X is located at the positionof ∠OIX = π/2.

Suppose that we now consider the locus of an arbitrary point on the peripheral surfaceof a rigid wheel when the wheel is running during driving action.

Figure 2.5 shows the rolling locus of an arbitrary point F on the peripheral surface ofthe rigid wheel when V = Rω i.e. the slip ratio id equals zero percentage. For a radius R,a moving velocity V and an angular velocity ω of the rigid wheel, the angle of rotation α

of the point F is given as ωt for an arbitrary time t. During the rotation, the center of thewheel O moves to O′ for the distance of Vt = Rα. The X , Y coordinates of the point F can


Figure 2.5. Cycloid curve for V = Rω.

then be expressed as:

X = R(α + sin α)

Y = R(1 + cos α) (2.20)

These equations are those of a cycloid curve and the rolling locus follows a cycloid path,where the coordinates of the point P at α = π/2 is X = R(π/2 + 1), Y = R. Likewise thecoordinate of the point Q at α = π is X = πR, Y = 0. Similarly, the coordinates of the pointS at α = 3π/2 is X = R(3π/2 − 1), Y = R and the coordinate of the point T of α = 2π isX = 2πR, Y = 2R.

The gradient of the tangent to the rolling locus dY /dX is coincident with the resultantvelocity vector of the soil particle on the peripheral surface of a rigid wheel as follows:

dY

dX= − sin α

1 + cos α= − tan

α

2(2.21)

The gradient of the tangent to the rolling locus calculates to −1 at the point P, −∞ at thepoint Q and then +1 and 0 at the point S and T, respectively.

Next, consider the problem of tracking the locus of an arbitrary point on the peripheralsurface of a rigid wheel when the wheel is rolling at a slip ratio id during driving action.Figure 2.6 shows the rolling locus of an arbitrary point F on the peripheral surface of a rigidwheel running in the condition of Rω > V . For an angle of rotation α at an arbitrary time t,α equals ωt and the center O of the wheel moves to O′ for a distance Vt = Vα/ω. Then, theX , Y coordinates of the point F can be expressed as:

X = Vα

ω+ R sin α = R {α(1 − id) + sin α} (2.22)

Y = R(1 + cos α) (2.23)

These mathematical expressions are those of a trochoid curve. Therefore, the rolling locusof a point on a slipping wheel follows a trochoid shaped path, where the coordinates of apoint P at α = π/2 are X = R{(1 − id)π/2 + 1} and Y = R. Similarly, the coordinates of thepoint Q at α = π are X = πR(1 − id) and Y = 0.

Likewise, the coordinates of the point S at α = 3π/2 are X = R{(1 − id)3π/2 − 1}, Y = Rand the coordinate of the point T at α = 2π are X = 2πR(1 − id), Y = 2R.

44 Terramechanics

Figure 2.6. Trochoid curve during driving action (Rω > V ).

Figure 2.7. Moving locus of a soil particle under rigid wheel during driving action [17] where xrepresents horizontal displacement and z represents vertical displacement.

The gradient of the tangent to the rolling locus dY /dX is also coincident with the resultantvelocity vector of the soil particle on the peripheral surface of a rigid wheel and may beexpressed as follows:

dY

dX= − sin α

1 − id + cos α(2.24)

The gradient of the tangent to the rolling locus is calculable as −1/(1 − id) at the point P, 0at the point Q, and +1/(1 − id) and 0 at the points S and T respectively. From Figure 2.6,it can be seen that the rolling locus draws a small loop around the contact part of the rigidwheel and the terrain which is a function of the amount of sinkage s. Also it can be seenthat a moving loss of 2πRid occurs due to the slippage during one revolution of the wheel.Yong et al. [16] used this rolling locus of a wheel as a boundary condition to an FEMstudy that they used to analyse the trafficability of a rigid wheel running on weak clayeyterrain.

Yong et al. [17] also observed the moving locus of the soil particles in a soil bin whena rigid wheel was running during driving action. Figure 2.7 shows the results of observationof the moving locus of the soil particle at a depth of 1.27 cm for various values of slip ratio id


Figure 2.8. Ground reaction acting on rigid wheel during driving action.

when a rubber coated rigid wheel of weight 240 N was driven on a clayey soil. The movinglocus of the soil particle show an elliptical path for each slip ratio.

Each soil particle moves forward and slightly upward during the progression of the wheel.It goes downward to a minimum position and then returns upward to a final position duringthe procession of the wheel. With increasing slip ratio id , the vertical distance of the finalposition of the soil particle i.e. the final amount of sinkage increases due to the increasingamount of slip sinkage and the horizontal distance of the final position tends to decrease.

2.2.3 Force balances

The traffic pattern of a rigid wheel when it is running during driving action at a constantmoving speed V and a constant angular velocity ω can exist in three, quite distinct, modesor regimes depending on the magnitude of the driving torque Qd > 0 that is applied to thewheel. The three operating regimes or modes of action are as shown in Figure 2.8.

Figure 2.8(a) shows that in mode 1 the effective driving force Td occurs in the movingdirection of the wheel when the driving torque Qd is comparatively small. The effectivedriving force Td balances with the horizontal ground reaction Bd which acts reversely to themoving direction of the wheel. The axle load W balances with the vertical ground reactionN . The direction of the resultant ground reaction, which is composed of N and Bd , does notgo through the wheel axle, but deviates a little bit to the front-side of the wheel axle. Theposition of the point of application of the ground reaction on the peripheral surface of therigid wheel has a horizontal position equal to the amount of eccentricity ed and is locatedat a vertical distance from the axle of ld .

The force balances in the horizontal and vertical directions, and the moment balancearound the wheel axle can be established as follows:

W = N

Td = Bd = µdW

Qd = Ned − Bdld (2.25)

46 Terramechanics

Then, the horizontal amount of eccentricity ed can be calculated for a coefficient of drivingresistance µd as,

ed = Qd

W+ µd ld (2.26)

Figure 2.8(b) shows a second mode of action where a rigid wheel is running in a self-propelling state. In this case, the effective driving force Td and the horizontal groundreaction Bd decrease with an increase in driving torque Qd . When Td and Bd become zero,only the vertical ground reaction acts on the wheel. At this time, the force and momentbalances and the horizontal amount of eccentricity ed can be established as follows:

W = N

Td = Bd = 0

Qd = Ned (2.27)

ed = Qd

W(2.28)

Figure 2.8(c) shows that, in a third mode of action, the effective driving force Td occursreversely to the moving direction of the wheel, when the driving torque Qd increases bya very large degree. Sufficient effective tractive effort to be able to draw another wheel(say) will develop when the driving force can overcome the land locomotion resistance ofthe wheel. The direction of the horizontal ground reaction Bd is coincident with the movingdirection of the wheel. The direction of the resultant ground reaction composed of N andBd deviates to a large extent to the front-side of the wheel axle.

The force and moment balances and the horizontal amount of eccentricity ed in this casecan be established as follows:

W = N

Td = Bd = µdW

Qd = Ned + Bdld (2.29)

ed = Qd

W− µd ld (2.30)

For each of these three traffic patterns, the actual effective driving force Td equals thehorizontal ground reaction Bd and can be calculated as the difference between the horizontalcomponent of the driving force (Qd /R)h acting forward to the moving direction of the wheeland the land locomotion resistance Lcd acting backward to the moving direction. Thisrelation can be expressed as follows:

Td =(

Qd

R

)h

− Lcd (2.31)

In interpreting this expression, it can be seen firstly that the actual horizontal groundreaction Bd and its direction depends on the slip ratio id and secondly that Bd equals thedifference between the driving force and the land locomotion resistance.


Figure 2.9. Distribution of contact pressure under rigid wheel during driving action.

2.2.4 Driving force

As shown in Figure 2.9, the distribution of the ground reaction applied to the peripheralsurface of a rigid wheel during driving action can be expressed by the positive distributionof normal stress σ and shear resistance τ around the contact part

�

A�

E, where A is thebeginning point of contact to the terrain, and E is the ending point of departure from theterrain. The resultant stress p applies to the peripheral surface at an angle of δ = tan−1(τ/σ)to the normal line. For an entry angle θf , exit angle θr and central angle θ between thevertical line OM and radius vector OX at an arbitrary point X on the peripheral surface ofthe rigid wheel, the amount of sinkage z for the central angle θ, the amount of sinkage s0

at the bottom-dead-center M, and the amount of rebound u0 after the pass of wheel can beexpressed respectively as follows:

z = R(cos θ − cos θf ) (2.32)

s0 = R(1 − cos θf ) (2.33)

u0 = R(1 − cos θr) (2.34)

The rolling locus of an arbitrary point X on the peripheral surface of the rigid wheel canbe determined using Eq. (2.22) and Eq. (2.23).

The length of trajectory l can be obtained by integrating elemental sections of the rollinglocus from X = a to b as in the following formula:

l =∫ b

a

√1 + (dY /dX )2dX

48 Terramechanics

Figure 2.10. Component of a rolling locus in the direction of applied stress during driving action.

Substituting X , Y and dX /dY given in Eqs. (2.22), (2.23) and (2.24) into the above equation,the length of trajectory l can be expressed as l(α) which is a function of angle α, as follows:

l(α) =∫ α

αf

√1 +

(− sin α

1 − id + cos α

)2

· R(1 − id + cos α) dα

Substituting the relation θ = π − α and θf = π − αf into the above equation, the followingfurther expression can be obtained:

l(θ) = R∫ θf

θ

{(1 − id)2 − 2(1 − id) cos θ + 1

} 12 dθ (2.35)

As shown in Figure 2.10, the direction of the resultant force between the effective drivingforce Td and the axle load W is given by the angle ζ = tan−1(Td /W ) to the vertical axis.In this case, the relationship between contact pressure q(θ) and soil deformation d(θ) inthe direction of the resultant force agrees well with the plate loading test result previouslygiven in Eq. (1.19). Here, q(θ) is the component of the resultant applied stress p(θ) to thedirection of the angle ζ to vertical axis. d(θ) is the component of the rolling locus l(θ) tothe same direction of the resultant force. That is, d(θ) is the component of the length oftrajectory l(θ) in the direction of applied stress q(θ). XT is an elemental length of trajectoryof l(θ) directed in the same direction as the resultant velocity vector of the vehicle velocityV and the circumferential speed Rω. Then, d(θ) can be calculated as the integral of XHfrom θ = θ to θf , which is the component of XT in the direction of the angle ζ to vertical


axis, as follows:

d(θ) = R∫ θf

θ

{(1 − id)2 − 2(1 − id) cos θ + 1

} 12

× sin{θ + ζ + tan−1 (1 − id) sin θ

1 − (1 − id) cos θ

}dθ (2.36)

The velocity vector Vp in the direction of the stress q(θ) at an arbitrary point X on thecontact part of the peripheral surface of a rigid wheel and the terrain can be calculated asthe component of the resultant velocity vector between V and Rω, as follows:

Vp = {(Rω − V cos θ)2 + (V sin θ)2} 1

2

× sin{θ + ζ + tan−1 (1 − id) sin θ

1 − (1 − id) cos θ

}(2.37)

Further, the plate loading and unloading test should be executed considering the ‘size effect’of the contact length of wheel b = R(sin θf + sin θr) and the ‘velocity effect’ of the loadingspeed Vp. The relationship between the stress component q(θ) of the resultant applied stressp(θ) and the soil deformation d(θ) can be determined as follows:

For θmax ≤ θ ≤ θf

q(θ) = p(θ) cos(ζ + θ − δ)

= k1ξ {d(θ)}n1

For −θr ≤ θ ≤ θmax

q(θ) = p(θ) cos(ζ + θ − δ)

= k1ξ {d(θmax)}n1 − k2 {d(θmax) − d(θ)}n2 (2.38)

ξ = 1 + λV κp

1 + λV κ0

k1 = kc1

b+ kφ1 and k2 = kc2

b+ kφ2

where θmax is the central angle corresponding to the maximum stress qmax = q(θmax). Thecoefficients kc1, kφ1 and kc2, kφ2, and the indices n1 and n2 need to be determined from thequasi-static plate loading and unloading test for a loading speed V0. The other coefficientλ and the index κ in the above coefficient of modification ξ for the previous Eq. (1.19)should be determined from a dynamic plate loading test at a plate loading speed of Vp.

Thence, the distribution of normal stress σ(θ) can be calculated as follows:

σ(θ) = p(θ) cos δ (2.39)

Also, the distribution of shear resistance τ(θ) may be calculated by substituting theamount of slippage jd given in Eq. (2.16) into the Janosi-Hanamoto’s equation (2.10),

50 Terramechanics

as follows:

τ(θ) = {ca + σ(θ) tan φ}× [

1 − exp{−aR

[(θf − θ) − (1 − id)(sin θf − sin θ)

]}](2.40)

The Eq. (2.40) can be applied for a loose accumulated sandy soil or weak clayey terrain,but another equation [12] can be used for hard compacted sandy terrains and so on.

Following this, the angle δ(θ) between the resultant applied stress p(θ) and the radialdirection can be calculated as follows:

δ(θ) = tan−1

{τ(θ)

σ(θ)

}(2.41)

Next, the axle load W , the driving torque Qd and the apparent effective driving force Td0

can be coupled (using the force balance relations) to the contact pressure distribution σ(θ)and τ(θ) as:

W = BR∫ θf

−θr{σ(θ) cos θ + τ(θ) sin θ} dθ (2.42)

Qd = BR2∫ θf

−θrτ(θ) dθ (2.43)

Td0 = BR∫ θf

−θr{τ(θ) cos θ − σ(θ) sin θ} dθ (2.44)

and then the driving force can be calculated as the value of Qd /R.The apparent effective driving force Td0 as shown in the above equation can be expressed

as a difference between thrust Thd and compaction resistance Rcd in the following equation:

Td0 = Thd − Rcd (2.45)

where

Thd = BR∫ θf

−θrτ(θ) cos θ dθ (2.46)

Rcd = BR∫ θf

−θrσ(θ) sin θ dθ (2.47)

In this case, the amount of slip sinkage ss is not considered. The thrust Thd can be calculatedas the integration of τ(θ) cos θ. It acts in the moving direction of the rigid wheel. On theother hand, the compaction resistance Rcd can be calculated as the integration of σ(θ) sin θ.This force acts reversely to the moving direction of the wheel and manifests as the landlocomotion resistance associated with the static amount of sinkage s0.

As shown in the diagram, the point Y on the peripheral surface of the rigid wheel isdetermined when the direction of the resultant applied stress p becomes vertical. Henceit can be seen that the horizontal ground reaction acting on the section AY of the contactpart to the terrain develops as the land locomotion resistance, whilst the horizontal groundreaction acting on the section YE develops as the thrust of the rigid wheel.


Figure 2.11. Rolling motion of a driven rigid wheel for a point X on the ground surface.

Additionally, the horizontal component of the driving force (Qd /R)h is given as the summa-tion of the actual effective driving force Td and the total compaction resistance Lcd – whichis the land locomotion resistance calculated considering the amount of slip sinkage.

2.2.5 Compaction resistance

In general, a considerable amount of slip sinkage will occur under a driven rigid wheel dueto the existence of the dilatancy phenomena [18] associated with the shear action of soil ata peripheral interface.

From a plate traction test, the amount of slip sinkage ss can be expressed as a functionof contact pressure p and amount of slippage js as follows:

ss = copc1 jc2s (2.48)

where the coefficient c0 and the indices c1, c2 are the terrain-wheel system constants. Thesevalues have to be determined experimentally for a given steel plate and terrain.

Figure 2.11 shows the rolling motion of a rigid wheel during driving action as it movesaround a point X on a terrain surface. To calculate the amount of slip sinkage ss at the pointX , it is necessary to determine the amount of slippage js at the point X .

When an arbitrary point F on the peripheral surface of the rigid wheel meets the point Aon the terrain surface at the time t = 0, the central angle of radius vector OF to the verticalaxis can be defined as the entry angle θf . Then, the central angle of the radial vector OF′to the vertical axis becomes θ at an arbitrary time t. The rotation angle ωt of the wheel canthen be expressed as:

ωt = θf − θ (2.49)

Thence the moving distance OO′ of the wheel during the time t can be calculated as:

OO′ = Vt = Rωt(1 − id) = R(1 − id)(θf − θ) (2.50)

52 Terramechanics

The transit time te, as the rigid wheel passes over the point X , may be taken as the timewhen the distance OO′′ reaches the contact length of wheel OO

′′ = R(sin θf + sin θr). Thecentral angle of the radius vector O′′F then becomes θe where:

θe = θf − sin θf + sin θr

1 − id(2.51)

Substituting the above angle θ = θe into Eq. (2.49) we get:

te = sin θf + sin θr

ω(1 − id)(2.52)

If we now recall, from the previous Section 2.2.2, that the amount of slippage of a rigidwheel during one revolution is 2πRid , then, the amount of slippage of soil js at the point Xmay be calculated as the ratio of te to 2π/ω as follows:

js = 2πRid · R(sin θf + sin θr)

2πR(1 − id)= R(sin θf + sin θr)

id1 − id

(2.53)

If we now take a small interval of central angle θn and a small interval of amount of slippagejs/N for a very small time interval te/N , then the vertical component of the resultant appliedstress p can be calculated as p(θn)cos(θn − δn) for the nth interval of the center angle θn.Then, by substituting these values into Eq. (2.48), the total amount of slip sinkage ss canbe calculated as the sum of the elemental amounts of slip sinkage as follows:

ss = c0

N∑n=1

{p(θn) cos(θn − δn)}c1

{( n

Njs)c2 −

(n − 1

Njs

)c2}

(2.54)

where

θn = n

N(θf + θr)

Consequent to this calculation, a rut depth i.e. the total amount of sinkage of a rigid wheel scan be calculated from the static amount of sinkage s0 given in Eq. (2.33). Also, the amountof rebound u0 given in Eq. (2.34) and the above mentioned amount of slip sinkage ss canbe determined as follows:

s = s0 − u0 + ss (2.55)

Next, the total compaction resistance Lcd can be calculated if we make the assumption thatthe product of the compaction resistance Lcd applied in front of the rigid wheel and themoving distance 2πR(1 − id) during one revolution of the wheel can be equated to the rutmaking work which in turn may be calculated as the integration of the contact pressure pacting on a plate of length 2πR(1 − id) and width B from an initial depth z = 0 to a totalamount of sinkage z = s,

2πR(1 − id)Lcd = 2πR(1 − id)B∫ s

0pdz


then, the total compaction resistance Lcd can be determined considering the vertical velocityeffect, by the following expression:

Lcd = B∫ s

0k1ξzn1 dz (2.56)

ξ = 1 + λV KZ

1 + λV K0

Vz = Rω sin{

cos−1

(1 − s − z

R

)}

where Vz is the vertical speed of the plate at depth z, and a modification coefficient ξ isused to take into account the velocity effect of the terrain-wheel system constants k1.

In the expression, Lcd is the value of the total land locomotion resistance of the rigid wheel.Consequently, the difference between Lcd and Rcd , which is the compaction resistance forthe static amount of sinkage s0−u0 given in Eq. (2.47), can be considered as the compactionenergy due to the amount of slip sinkage ss.

2.2.6 Effective driving force

As shown in Figure 2.8, the effective driving force Td and the axle load W act on the centralaxis of a rigid wheel whilst the horizontal ground reaction Bd and the vertical reaction forceN act on a point deviated from the bottom-dead-center by an eccentricity amount ed and bya vertical distance ld from the wheel axle. From force balance and horizontal and verticalequilibrium considerations, the effective driving force Td and the axle load W must equalthe horizontal reaction force Bd and the vertical component N respectively. Td can now becalculated as the difference between the horizontal component of the driving force (Qd /R)h

and the compaction resistance Lcd , as given in the following equation:

Td = Bd =(

Qd

R

)h

− Lcd (2.57)

The amount of eccentricity ed0 for the no slip sinkage state can be worked out from consid-erations of the moment equilibrium of the vertical stress applied to the peripheral contactsurface taken around the axle of the rigid wheel. Taking moments, the product of Nand ed0 equals the integral of the product of R sin θ and the vertical component of appliedstress pRB cos θ cos(θ−d)dθ acting on an element of contact area RB dθ cos θ at an arbitrarypoint X on the peripheral surface of the rigid wheel. The result of doing this is:

Ned0 = BR2∫ θf

−θr

p cos θ cos(θ − δ) sin θ dθ

Substituting N = W into the above equation, we get:

ed0 = BR2

W

∫ θf

−θr

p cos(θ − δ) sin θ cos θ dθ (2.58)

54 Terramechanics

Thence, the real eccentricity ed of the vertical reaction force N can be modified as follows,taking into consideration the amount of slip sinkage:

where

θ′f = cos−1

[1 − s

R

]and θ′

r = θr

and

ld = Wed

Lcd

Using these expressions the position of the ground reaction force ed and the value of ld canbe determined.

2.2.7 Energy equilibrium

Following the principle of conservation of energy, it is evident that the effective inputenergy E1 supplied by the driving torque Qd to the rigid wheel is equal to the sum ofthe individual output energy components. These components are the sinkage deformationenergy E2 required to make a rut under the rigid wheel, the slippage energy E3 whichdevelops at the peripheral contact part of the wheel and the effective drawbar pull energyE4 which is required to develop an effective driving force. Energy balance requires that:

E1 = E2 + E3 + E4 (2.59)

In the general case, the energy developed during the rotation of the radius vector from theentry angle θf to the central angle θf + θr is calculable. As the rigid wheel rotates R(θf + θr)and experiences a moving distance of R(θf + θr)(1 − id) during this rotation, the variousindividual energy components can be computed as follows:

E1 = Qd(θf + θr) (2.60)

E2 = LcdR(θf + θr)(1 − id) (2.61)

E3 = Qd(θf + θr)id (2.62)

E4 = TdR(θf + θr)(1 − id) (2.63)

Next, one can note that the, value of the amount of energy developed per second can beexpressed in terms of the peripheral speed Rω and the moving speed V of the rigid wheel,as follows:

E1 = Qdω =(

Qd

R

)h

V

1 − id(2.64)

E2 = LcdRω(1 − id) = LcdV (2.65)

E3 = Qdωid =(

Qd

R

)h

idV

1 − id(2.66)

E4 = TdRω(1 − id) = TdV (2.67)

The optimum effective driving force Tdopt during driving action is defined as the effectivedriving force at the optimum slip ratio iopt . This occurs when the effective drawbar pull


energy E4 takes a maximum value under a constant peripheral speed Rω. The tractive powerefficiency Ed is defined by the following equation:

Ed = Td(1 − id)

[Qd/R]h(2.68)

2.3 AT BRAKING STATE

2.3.1 Amount of slippage

Figure 2.12 shows the distribution of the slip velocity and the amount of slippage aroundthe peripheral surface of a rigid wheel during braking action. The skid ib during brakingaction is defined for Rω < V as follows:

ib = Rω

V− 1 (2.69)

The tangential slip velocity Vs at an arbitrary point X corresponding to a central angle θ

on the peripheral contact part of a rigid wheel is determined as follows. For a radius R, anangular velocity ω = (−dθ/dt), and a moving speed of the rigid wheel V in the direction ofthe terrain surface the following relationship can be established.

Vs = Rω − V cos θ

Vs = Rω

(1 − 1

1 + ibcos θ

)(2.70)

where θ is the angle between OM and radius vector OX and is positive for the counter-clockwise direction. Vs becomes zero at the point X0 on the peripheral contact surface andthe central angle,

θ = cos−1

(Rω

V

)= cos−1(1 − ib)

Figure 2.12. Distribution of slip velocity Vs and amount of slippage jb during braking action (Rω < V ).

56 Terramechanics

and the slip velocity take on positive values at the contact section X0A and take on negativevalues at the contact section X0E.

When ib = cos θf − 1 i.e. X0 becomes coincident with the point A of the entry angle θf ,Vs becomes negative for the whole range of contact part AE.

The amount of slippage jb(θ) is shown as the integration of Vs from the initial time t = 0and a central angle θ = θf at the beginning of contact with the soil to an arbitrary time t = tand a central angle θ = θ. Thence, the following equation is obtained.

jb = R∫ t

0ω

(1 − 1

1 + ibcos θ

)dt = R

∫ θf

θ

(1 − 1

1 + ibcos θ

)dθ

= R

{(θf − θ) − 1

1 + ib(sin θf − sin θ)

}(2.71)

For the range of cos ωf − 1 < ib < 0, jb(θ) takes the maximum value jb(θm) at the pointX0 and at the central angle θm = cos−1(Rω/V ), where Vs becomes zero. And jb(θ) becomeszero at the point C and at the central angle θ = α0 which is derived from the followingequation:

(θf − θ)(1 + ib) = sin θf − sin θ (2.72)

That is, the integrated area of Vs from the point A to X0 is equal to the area of Vs integratedfrom the point X0 to C.

The shear resistance τ(θ) takes a positive value for the loading state while the amount ofslippage jb(θ) increases to a maximum value jb(θm) and for the unloading state, while jb(θ)decreases to some value jb(θr) corresponding to zero shear resistance. Similarly τ(θ) takesa negative value for the reverse traction state when jb(θ) becomes less than jb(θr). On theother hand, for the range of ib ≤ cos θf − 1, jb(θ) takes a negative value and τ(θ) also takesa negative value for the whole range of the central angle of −θr < θ < θf .

2.3.2 Soil deformation

Figure 2.13 shows the general flow pattern beneath a towed rigid wheel in sandy soil andthe position of an instantaneous center. The slip lines are largely divided into two parts forleft and right hand sides from the point N on the peripheral contact area.

The forward and backward flow zones NCA and NDE are bounded by a logarithmicspiral for the transient shear zone from active to passive state and then a straight line whichmeets the ground surface at the angle π/4 − φ/2 rad at the points C and D, respectively, forthe passive failure zone. The parameter φ is the angle of internal friction of the soil.

As the instantaneous center of the towed wheel I is situated below the bottom-dead-center M, section

�

N�

A, of the rim has a generally forward and downward movement andmoves the soil upwards in the region NCA and therefore the soil particles slide along therim in such a way as to produce shear resistance in the direction opposite to that of wheelrevolution. Similarly section

�

N�

E of the rim has a relatively fast forward movement whilethe soil particles along the rim move forward slowly and therefore the shear resistance actsin the direction of wheel revolution. In general, for a towed rigid wheel, the shear resistancechanges its direction at the point N on the peripheral contact surface, which is called the


Figure 2.13. Soil deformation under a towed rigid wheel, instantaneous center I, and directions ofslip line S1, S2 and principal stress σ1, σ3.

transition point. Wong et al. [19] verified experimentally for the behaviour of sandy soilbeneath a towed rigid wheel – that the transition point N corresponds to the position wherethe two flow zones meet each other, as shown in the diagram, and the location of thetransition point corresponds to the point C in the previous Figure 2.12. Point C is wherethe shear resistance τ becomes zero and the normal stress σ becomes the major principalstress at the interface of the soil mass. Two conjugate slip lines [20] s1, s2 occur forwardand backward beneath the point N and intersect each other at an angle ±(π/4 − φ/2) radto the direction of the maximum principal stress σ1 and at an angle ±(π/4 + φ/2) rad tothe direction of the minimum principal stress σ3.

The resultant velocity vector of all the soil particles on the peripheral contact part of therigid wheel always rotates around the instantaneous center I. The position of I lies belowthe bottom-dead-center M at a distance OI from the axle of the wheel. Using the notationshown on the diagram, the distance OI can be calculated as:

OI = R cos θ + R sin θ tan(θ + θn) = R cos θ + 1

ω(V − Rω cos θ)

= V

ω= R

1 + ib(2.73)

58 Terramechanics

The next equation can be derived for the triangle OXI in this diagram as,

OI

sin(π/2 + θn)= R

sin{π/2 − (θ + θn)}therefore

OI

R= cos θn

cos(θ + θn)

and then, a substitution of the above equation into Eq. (2.73) gives:

1 + ib = cos θ − sin θ tan θn

tan θn = cos θ − (1 + ib)

sin θ(2.74)

Wong [21] concluded that the central angle θ = θn corresponding to the point N could bedetermined by substituting θn = π/4 − ψ/2 into the above equation as follows:

tan(

π

4− ψ

2

)= cos θtr − (1 + ib)

sin θtr(2.75)

From the above Eq. (2.73), the instantaneous center I lies on the bottom-dead-center M atib = 0% and moves its location deeper below the point M with increments of skid |ib|. So,the position of instantaneous center I depends on the skid ib, or the moving speed V andthe angular velocity ω of the wheel. As can be simply shown, the angle θn between theresultant velocity vector and the radial direction at point X becomes zero at ∠OXI = π/2and at θ = cos−1(1 + ib).

As will be mentioned later, the ground reaction acting on the contact part of a rigidwheel

�

A�

N develops the land locomotion resistance while another ground reaction actingon the contact part

�

N�

E develops the drag for the rigid wheel. These ground reactions tothe rigid wheel apply reversely to the moving direction, and they will make the rigid wheelbrake.

When a rigid wheel is locked at a skid ib = −100%, the point N approaches the pointE so that the drag disappears and the slip line develops only on the curve

�

N�

C. The landlocomotion resistance acts as a form of bulldozing resistance to the rigid wheel.

Next, the trajectories i.e. the rolling locus of a point on the peripheral contact part of therigid wheel during braking action will be considered. Figure 2.14 shows the rolling locus

Figure 2.14. Trochoid curve during braking action (V > Rω).


of an arbitrary point F on the peripheral surface of a rigid wheel for a skid ib and Rω < Vduring braking action. Given that the relation between the central angle α of the point F andthe time t can be expressed as α = ωt, the position of the axle of the rigid wheel O movesto O′ in a distance Vt = Vα/ω.

Thence, the coordinates (X ,Y ) can be calculated as follows:

X = Vα

ω+ R sin α = R

(α

1 + ib+ sin a

)(2.76)

Y = R(1 + cos α) (2.77)

The above equations imply a trochoid curve without singular points. The coordinates ofpoint P at α = π/2 are calculated as X = R[π/{2(1 + ib)}+ 1] and Y = R, the coordinates ofQ at α = π are calculated as X = R{π/(1 + ib)} and Y = 0, the coordinates of S at α = 3π/2are calculated as X = R[3π/{2(1 + ib)} − 1] and Y = R, and the coordinates of T at α = 2π

are calculated as X = 2πR/(1 + ib) and Y = 2R. The gradient of the tangent to the rollinglocus dY /dX is equal to the direction of the resultant velocity vector of an arbitrary point onthe peripheral contact surface of the rigid wheel. This gradient can be expressed as follows:

dY

dX= − sin α

1/(1 + ib) + cos α(2.78)

Then, the gradients of the tangents to the rolling locus at point P, Q, S and T can bedetermined as −(1 + ib), 0, +(1 + ib), and 0, respectively. During one revolution of a rigidwheel, there is a loss −2πR ib/(1 + ib) in the wheel advance and the moving distance isexpressed as 2πR/(1 + ib) in conjunction with an amount of sinkage s.

Wong [15] observed the trajectories of the soil particles in a soil bin, filled with clay,when a rigid wheel was rotating during braking action.

As to his results, it was observed that the final horizontal amount of movement of a soilparticle to the moving direction of the wheel decreased with depth. A soil particle locatedat a shallow depth in the soil bin moved first along a π/4 rad line upward to the horizontalline when the wheel approached the point. Then, the soil particle moved downward alonga circular trajectory. After the pass of the wheel, the soil particle moved upward and reachedits initial position due to a rebound action in the non-compressible saturated clay.

2.3.3 Force balances

The traffic pattern of a rigid wheel which is running during braking action at a constantmoving speed V and at a constant angular velocity ω can be divided into two quite distinctiveregimes depending on the amount of braking torque Qb (≤0). One regime is the pure rollingstate at Qb = 0 and the other is the braking state at Qb < 0.

Figure 2.15(a) shows a rigid wheel which is running in a pure rolling mode. In thiscase, i.e. when the braking torque becomes zero, an effective braking force Tb = Tr occursopposite to the moving direction of the wheel. Tr is called the ‘rolling resistance’. It balanceswith the horizontal ground reaction Bb = Br which acts reversely to the moving direction ofthe wheel. Since the axle load W balances with the vertical ground reaction N , the directionof the resultant ground reaction (which is composed of N and Bb = Br) goes through thewheel axle. The position of the application of the net ground reaction on the peripheral

60 Terramechanics

surface acts at a point on the wheel with an horizontal amount of eccentricity eb = er anda vertical distance from the axle lb = lr , as shown in the diagram. By equating forces inthe horizontal and vertical directions, and by equating moments around the wheel axle thefollowing equations may be derived:

W = N

Tr = Br = µrW

−Ner + Brlr = 0 (2.79)

Then, the horizontal amount eccentricity er can be calculated for a coefficient of rollingresistance µr as follows:

er = µrlr (2.80)

In a similar vein, Figure 2.15(b) shows a rigid wheel that is running in a braking state. Inthis case, the effective braking force Tb which is directed in the moving direction of thewheel increases when a braking torque Qb is applied. Tb can also do external work as anactual braking force to another wheel. The horizontal ground reaction Bb acts oppositely tothe moving direction of the wheel. The direction of the resultant ground reaction composedof N and Bb deviates to the left hand side of the wheel axle.

In this case, the horizontal and vertical force balances and the moment balance about theaxles yield the following results:

W = N

Tb = Bb = µbW

Qb = −Neb + Bblb (2.81)

The value of the horizontal amount of eccentricity eb can be calculated for a coefficient ofbraking resistance µb as:

eb = −Qb

W+ µblb (2.82)

Figure 2.15. Ground reaction acting on rigid wheel during braking action.


In this case, the actual effective braking force Tb equals the horizontal ground reactionBb, which is given as the difference between the horizontal component of the braking force(Qb/R)h acting reversely to the moving direction of the wheel and the compaction resistanceLcb, as follows:

Tb =(

Qb

R

)h

− Lcb (2.83)

The horizontal ground reaction Bb will be determined by both these values depending onthe skid ib.

2.3.4 Braking force

Figure 2.16 shows the contact pressure distributions that apply on the peripheral contactsurface of a rigid wheel during braking action. The normal stresses σ(θ) have positive valuesfor the entire contact portion of the rigid wheel. On the other hand, the shear resistances τ(θ)applied on the contact section

�

A�

X0 have positive values in correspondence with positiveamounts of slippage jb(θ), but τ(θ) applied on the contact section

�

X0�

E0 turn to negativevalues in correspondence with negative amounts of slippage jb(θ). The angles δ(θ) betweenthe resultant applied stress p(θ) and the radial direction of the wheel surface are expressed as:

δ(θ) = tan−1

{τ(θ)

σ(θ)

}These are positive in section

�

A�

X0, but negative in section�

X0�

E.

Figure 2.16. Contact pressure distributions applied on peripheral surface of rigid wheel duringbraking action.

62 Terramechanics

Figure 2.17. Component of a rolling locus �d(θ) in the direction of applied stress during brakingaction.

At the point X0 at which the shear resistance τ(θ) and the slip velocity Vs become zero, andat which the amount of slippage jb takes a maximum value, the applied normal stress σ(θ)becomes the maximum principal stress σ1 and the direction of the resultant stress p = σ(θ)is coincident with the radial direction of the rigid wheel.

The amount of sinkage z at an arbitrary point X on the peripheral surface, the amount ofsinkage s0 at the bottom-dead-center M and the amount of rebound u0 at the point E canall be expressed as shown in Eqs. (2.32) ∼ (2.34).

The length of rolling locus l = l(α) can be derived from Eqs. (2.76) and (2.77). Sub-stituting the relation θ = π − α into l(α), the length of the trajectory l = l(θ) is given byintegrating the elemental locus from θ = θ to θ = θf as follows:

l(θ) = R∫ θf

θ

{(1

1 + ib

)2

− 2 cos θ

1 + ib+ 1

} 12

dθ (2.84)

As shown in Figure 2.17, the direction of the resultant force between the effective brakingforce Tb and the axle load W is given by the angle ζ = tan−1(Tb/W ) to the vertical axis.In this case, the relation between contact pressure q(θ) and soil deformation d(θ) in thedirection of the resultant force agrees well with the plate loading, unloading and sinkagerelationships. Here, q(θ) is the component of the resultant applied stress p(θ) to the directionof the angle ζ to the vertical axis. d(θ) is the component of the rolling locus l(θ) to thesame direction of the resultant force. As XT is the element of the rolling locus in the samedirection of the resultant velocity vector, d(θ) may be calculated as the integral of XHfrom θ = θ to θ = θf , which is the component of XT in the direction of the angle ζ to the


vertical axis, as follows:

d(θ) = R∫ θf

θ

{(1

1 + ib

)2

− 2 cos θ

1 + ib+ 1

} 12

× sin(θ − ς + tan−1 sin θ

1 + ib − cos θ

)dθ

(2.85)

η = tan−1 V sin θ

Rw − V cos θ= tan−1 sin θ

1 + ib − cos θ

ξ = δ + η − π/2, ∠HXT = η − (π/2 − θ + ζ)

�d(θ) = XH = XT cos(∠HXT ) = �l(θ) sin(θ − ζ + η)

q = p cos(ζ − θ + ζ)

The velocity vector Vp in the direction of the stress q(θ) applied to the peripheral surfaceof a rigid wheel can be calculated as the component of the resultant velocity between Rω

and V , as follows:

Vp = {(Rω − V cos θ)2 + (V sin θ)2} 1

2 × sin(θ − ς + tan−1 sin θ

1 + ib − cos θ

)(2.86)

In this case, the plate loading and unloading test should be carried out allowing for the ‘sizeeffect’ of the contact length of the wheel b = R(sin θf + sin θr) and the ‘velocity effect’ ofthe loading speed Vp. Thereafter, the relationship between the resultant applied stress p(θ),the stress component q(θ) and the soil deformation d(θ) can be determined as:

For θmax ≤ θ ≤ θf

p(θ) = k1ξ{d(θ)}n1

cos(ς − θ + δ)

For −θr < θ < θmax

p(θ) = k1ξ{d(θmax)}n1 − k2{d(θmax) − d(θ)}n2

cos(ς − θ + δ)(2.87)

ξ = 1 + λV kp

1 + λV k0

k1 = kc1

b+ kφ1 and k2 = kc2

b+ kφ2

where θmax is the central angle corresponding to the maximum stress qmax = q(θmax). Thecoefficients kc1, kφ1 and kc2, kφ2 and the indices n1 and n2 in this case need to determinedfrom the quasi-static plate loading and unloading test which is carried out at a loading speedV0. The other coefficients λ and the index κ need to be determined by use of a dynamicplate loading test carried out at a loading speed Vp.

Thence, the distribution of normal stress σ(θ) can be calculated as:

σ(θ) = p(θ) cos δ (2.88)

64 Terramechanics

Similarly, the distribution of shear resistance τ(θ) may be calculated by substituting theamount of slippage jb(θ) in Eq. (2.71) into the previously presented Janosi-Hanamotoequation, as follows:

For a first mode case where cos θf − 1 < ib < 0:

For a traction state of 0 < jb(θ) < jp,

τ(θ) = {ca + σ(θ) tan φ}[1 − exp{−ajb(θ)}]For a untraction state of jq < jb(θ) < jp,

τ(θ) = τp − k0{jp − jb(θ)}n0

For a reciprocal traction state of jb(θ) ≤ jq,

τ(θ) = −{ca + σ(θ) tan φ}(1 − exp[−a{jq − jb(θ)}]) (2.89)

where jp is the maximum amount of slippage, τp is the shear resistance at jp, and jq is theamount of slippage when τ takes on a value of zero during the untraction state.

For a first mode case where ib < cos θf − 1:

τ(θ) = −{ca + σ(θ) tan φ}[1 − exp{ajb(θ)}] (2.90)

The above equations can be applied for loose sandy soil or weak clayey soft ground, butanother equation [12] is better used in the case of hard compacted sandy ground.

Taking a next step, the axle load W , the braking torque Qb, and the apparent effectivebraking force Tb0 can be related to the contact pressure distribution σ(θ) and τ(θ), as follows:

W = BR∫ θf

−θr

{σ(θ) cos θ + τ(θ) sin θ} dθ (2.91)

Qb = BR2∫ θf

−θr

τ(θ) dθ (2.92)

Tbo = BR∫ θf

−θr

{τ(θ) cos θ − σ(θ) sin θ} dθ (2.93)

Thence, the braking force can be computed as the value of Qb/R.In the above equations, Qb ≤ 0 and Tb0 < 0, and the apparent effective braking force Tb0

can be expressed as the difference between the drag Thb(<0) and the compaction resistanceRcb(>0), as follows:

Tbo = Thb − Rcb (2.94)

where

Thb = BR∫ θf

−θr

τ(θ) cos θ dθ (2.95)

Rcb = BR∫ θf

−θr

σ(θ) sin θ dθ (2.96)


Further, the horizontal component of the braking force (Qb/R)h may be realised as the sumof the actual effective braking force Tb and the total compaction resistance Lcb which is theland locomotion resistance calculated for the amount of slip sinkage.


Figure 2.18 shows the rolling motion of a rigid wheel during braking action on a groundsurface.

To calculate the amount of slip sinkage ss of a wheel at a point X on the ground surface,it is necessary to determine the amount of slippage js at a point X . The moving distanceOO′ of the wheel during the time t is given from Eq. (2.49) as follows:

OO′ = Vt = Rωt

1 + ib= R(θf − θ)

1 + ib(2.97)

The transit time te during which a rigid wheel passes over a point X is given as the timewhen the distance OO′ reaches the contact length of wheel OO′′ and the central angle ofthe radius vector O′′F becomes θ = θe as follows:

OO′′ = R(sin θf + sin θr)

θe = θf − (1 + ib)(sin θf + sin θr) (2.98)

therefore

te = 1

ω(1 + ib)(sin θf + sin θr) (2.99)

is obtained.Since the amount of slippage of a rigid wheel during one revolution of the wheel is

equivalent to −2πRib/(1 + ib), the amount of slippage of a soft ground at a point X can be

Figure 2.18. Rolling motion of a towed rigid wheel for point X on ground surface.

66 Terramechanics

calculated as the ratio of te to 2π/ω as follows:

js = −2πRib1 + ib

· R(sin θf + sin θr)

2πR/(1 + ib)= −Rib(sin θf + sin θr) (2.100)

Considering a small interval of the central angle (θf + θr)/N and a small interval of theamount of slippage js/N for a very small time interval te/N , the vertical component ofthe resultant applied stress p can be calculated as p(θn) cos(θn − δn) for the nth interval ofthe central angle θn. Then, substituting this into Eq. (2.48), the amount of slip sinkage ss

can be developed as the sum of all the elemental amounts of slip sinkage as follows:

ss = c0

N∑n=1


{( n

Njs)c2 −

(n − 1

Njs

)c2}

(2.101)

where

θn = n

N(θf + θr)

Further, the total sinkage i.e. the rut depth of the rigid wheel s can be calculated given theamount of static sinkage s0, the amount of slip sinkage ss and the amount of rebound u0,as follows:

s = s0 − u0 + ss (2.102)

The product of the compaction resistance Lcb applied in front of the rigid wheel and themoving distance 2πR/(1+ ib) during one revolution of the wheel can be equated to the workin making the rut. This work component can be calculated as the integration of the contactpressure p acting on a plate of length 2πR/(1 + ib) and of B from a depth z = 0 to the totalamount of sinkage z = s, as follows:

2πRLcd

1 + ib= 2πRB

1 + ib

∫ s

0pdz

Thence the compaction resistance Lcb can be determined as follows, considering the verticalvelocity effect:

Lcb = k1ξB∫ s

0zn1 dz (2.103)

where

ξ = 1 + λV kz

1 + λV k0

Vz = Rω sin{

cos−1

(1 − s − z

R

)}

Additionally, Lcb is the value of the total land locomotion resistance to the rigid wheel’spassage. Hence, the difference between Lcb and Rcb – which is the compaction resistancefor the static amount of sinkage s0 − u0 given in Eq. (2.96) – can be considered as thecompaction resistance due to the amount of slip sinkage ss.


2.3.6 Effective braking force

As shown in Figure 2.15, the effective braking force Tb and the axle load W act on thecentral axis of a rigid wheel. Likewise, the horizontal ground reaction Bb and the verticalreaction N act on a point deviated from the bottom-dead-center by an eccentricity amounteb and by a vertical distance lb from the wheel axle. From force balance considerations,Tb and W equal Bb and N respectively. Tb can be calculated as the difference between thehorizontal component of the braking force (Qb/R)h and the compaction resistance Lcb, asin the following equation:

Tb = Bb =(

Qb

R

)h

− Lcb (2.104)

The amount of eccentricity eb0 for the no slip sinkage state is given as follows, consideringthe moment equilibrium of vertical stress applied to the peripheral contact surface aroundthe axle of the rigid wheel. That is, the product of N and eb0 equals the integral of the productof R sin θ and the vertical component of the applied stress times the elemental contact areaas predicted for the driving state.

Then, the amount of eccentricity eb0 can be calculated as follows:

eb0 = BR2

W

∫ θf

−θr

p cos(θ − δ) sin θ cos θ dθ

Following this, the real eccentricity eb of the vertical ground reaction N can be modified as:

eb = (sin θ′f + sin θ′

r)(R sin θr + eb0)

sin θf + sin θr− R sin θr (2.105)

θ′f = cos−1

[1 − s

R

]and θ′

r = θr

and

lb = Web

Lcb

Finally, the position of ground reaction force, eb and lb can be determined.

2.3.7 Energy equilibrium

Applying the conservation of energy principle, it is clear that the effective input energy E1

supplied by the braking torque Qb to a rigid wheel is equal to the sum of the individual outputenergy components. These components are the sinkage deformation energy E2 required tomake a rut under the rigid wheel, the slippage energy E3 which develops on the peripheralcontact part of the wheel and the effective braking force energy E4.

In this case, the energy developed during the rotation of the radius vector from the entryangle θf to the central angle θf + θr can be considered. As the rigid wheel rotates R(θf + θr)and the moving distance becomes R(θf + θr)/(1 + ib), each component energy factor can

68 Terramechanics

be quantitatively defined as follows:

E1 = Qb(θf + θr) (2.106)

E2 = LcbR(θf + θr)

1 + ib(2.107)

E3 = Qb(θf + θr)ib1 + ib

(2.108)

E4 = TbR(θf + θr)

1 + ib(2.109)

Next, one can note that the value of the amount of energy developed per second can beexpressed in terms of the peripheral speed Rω and the moving speed V of the wheel, asfollows:

E1 = Qbω =(

Qb

R

)h

V (1 + ib) (2.110)

E2 = LcbRω

1 + ib= LcbV (2.111)

E3 = Qbωib1 + ib

=(

Qb

R

)h

ibV (2.112)

E4 = TbRω

1 + ib= TbV (2.113)

The optimum effective braking force Tbopt is defined as the effective braking force at the opti-mum skid ibopt when the effective input energy |E1| takes a maximum value. Additionally,the braking power efficiency Eb can be defined as:

Eb = Tb

(Qb/R)h(1 + ib)(2.114)

2.4 SIMULATION ANALYSIS

Figure 2.19 presents a flow chart that can be used to calculate the tractive performance ofa driven rigid wheel or the braking performance of a towed rigid wheel when it is runningon a flat soft ground. First of all, the various wheel dimensions – such as the axle loadW , the radius R, the width B of the rigid wheel, the peripheral velocity Rω or the wheelvelocity V are required to be provided as a set of input data.

Following this, the terrain-wheel system constants – such as the coefficients kc1, kφ1, andkc2, kφ2 and the indices n1, n2, and the loading rate V0 measured from the quasi-static plateloading and unloading test, the coefficient α and the index κ measured from the dynamicplate loading test, the soil constants ca, tan φ and a measured from the plate traction test,as well as the coefficient c0, the indices c1, c2 measured from the slip sinkage test are alsorequired as another set of input data.

For the rest condition, the distributions of normal stress σ(θ), shear resistance τ(θ),resultant applied stress p(θ) and friction angle δ(θ) can be calculated for a given entry angle


W, B, R, V or Rω

kc1, kϕ1, n1, kc2, kϕ2, n2, Vo, λ, κ , co, tanϕ, a, co, c1, c2

pr(θ), σr(θ), τr(θ), δr(θ)

pf(θ), σf(θ), τf(θ), δf(θ)

|W–W(θf, θr)| < ε

si = soi + sst

Tdopt, idopt, Tbopt, ibopt, Tdmax, idmax, Tbmax, ibmax

θfoi, θr0i, p0i(θ), σ0i(θ), τ0i(θ), δ0i(θ), soi, uoj

θfo, θr0, po(θ), σo(θ), τo(θ), δo(θ), so

p(θ), σ(θ), τ(θ), δ(θ)

θfi, θri, σ(θ), τ(θ), si, ui

id or ib

Qdi or Qbi, Tdi or Tbi, Thdi or Thbi, Rcdi or Rcbi,

N N

NN

N

STOP

END

|θroi–θroij| < ε

|pf(θ) - pfj(θ)| < ε

|p(θ)–pj(θ)| < ε

θfoi

θfo

|W–W(θo)| < ε

Figure 2.19. Simulation flow chart.

θf 0 and for a given exit angle θr0. Thence, to determine the real values of θf 0 and θr0, thedistributions of σ(θ), τ(θ), p(θ) and δ(θ) can be iteratively calculated by means of the twodivision method until the vertical equilibrium Eq. (2.91) can be precisely satisfied.

At driving or braking state for a given slip ratio id or skid value ib, the following twocalculations need to be undertaken so that the entry angle θfoi may be determined. For theforward peripheral contact part

�

A�

M, the distribution of σf (θ) calculated from Eq. (2.39)or (2.88), τf (θ) calculated from Eq. (2.41) or (2.90), which can be calculated from pf (θ)given in Eq. (2.38) or (2.87) are calculated repeatedly until the real distribution of pf (θ)is determined. After that, for the backward peripheral contact part

�

M�

E, the distributionof σr(θ), τr(θ), and δr(θ) calculated from pr(θ) are computed until the exit angle θroi isdetermined.

70 Terramechanics

Next, calculations should be done for the given entry angle θfoi until the vertical equi-librium Eq. (2.42) or (2.91) are satisfied precisely for the given slip ratio id or skid valueib. Then, the final values of θfoi and θroi, the final distribution of pf (θ) and σf (θ), τf (θ),δf (θ), and pr(θ) and σr(θ), τr(θ), and δr(θ), the final amount of sinkage s0i and the finalamount of rebound u0 can be determined. Further to this, the total amount of sinkage si maybe calculated from Eq. (2.55) or (2.102) else can be determined from the amount of slipsinkage ssi given in Eq. (2.54) or (2.101). Then, the driving or braking torque Qdi or Qbi canbe calculated from Eq. (2.43) or (2.92), the driving force Qdi/R or the braking force Qbi/R,the compaction resistance Lcdi or Lcbi can be calculated using Eq. (2.56) or (2.103), theamount of eccentricity edi or ebi can be calculated from Eq. (2.58) or (2.105), the tractiveor braking power efficiency Edi or Ebi can be calculated from Eq. (2.68) or (2.114) andseveral energy values E1i , E2i , E3i and E4i can be calculated from Eq. (2.60) ∼ (2.67) or(2.106) ∼ (2.113) and can be determined for all the slip ratios id or skids id . Additionally,the optimum slip ratio idopt or the optimum skid ibopt and the optimum effective drivingor braking force Tdopt or Tbopt can be determined. Finally, the relations between Qd /R–id ,Td–id , s–id , θdf , θdr–id , ed–id , Ed–id and Qb/R–ib, Tb–ib, s–ib, θbf , θbr–ib, eb–ib, Eb–ib, and,E, E2, E3, E4–id or ib, and the distributions of the normal stress σ(θ) and the shear resistanceτ(θ) for all the slip ratios id or skids id can be graphically developed and portrayed by useof an ordinary microcomputer.

2.4.1 Driving state

To give an example of these computational processes and to validate them, the tractiveperformance of a driven rigid wheel with an axle load W = 1.52 kN, and geometrical prop-erties radius R = 16 cm and width B = 9.5 cm has been simulated for a wheel running ona weak soft soil ground with a peripheral speed Rω = 7.07 cm/s. The analytical simulationresults have then been contrasted and verified by comparison with detailed experimentaltest data. All the terrain-wheel system constants for the experimental situation are given inTable 2.1.

The sandy soil in the experiment was dried in air and had a water content w = 2.38%,a specific gravity Gs = 2.66, an average grain size D50 = 0.78 mm and a coefficient ofuniformity Uc = 12.0. The size of soil bin was 120 cm in length, 10 cm in width and 35 cmin depth. The sandy soil was filled uniformly into the two dimensional soil bin by meansof a free fall method that employed a 35 cm drop height. The initial density of the sandy

Table 2.1. Terrain-wheel system constants.

Plate loading and unloading testkc1 = 14.46 N/cmn1+1 kc2 = 48.10 N/cmn2+1

kφ1 = 4.95 N/cmn1+2 kφ = 36.47 N/cmn2+2

n1 = 0.809 n2 = 0.757λ = 0.18κ = 1.35 V0 = 0.035 cm/s

Plate traction and slip sinkage testca = 0 kPa tan φ = 0.423 a = 1.76 1/cmc0 = 9.738 × 10−6 c1 = 2.065 c2 = 1.074


soil was prepared to be γ = 1.52 mg/m3. The actual traction test on the rigid wheel wasexecuted under conditions of plane strain with the driving torque Qd , the effective drivingforce Td , and the total amount of sinkage s being measured directly.

Figure 2.20 shows the experimental relationships that exist between the driving forceQd /R, the effective driving force Td and the slip ratio id . Qd /R increases gradually withincrements in id and increases rapidly from id ∼= 55%. Td also increases gradually withid and reaches a maximum value Tdmax = 0.127 kN at idm = 21%. After that, Td takes anegative value at id = 43% and then decreases rapidly at slip ratios more than about 55%.As a consequence, the rigid wheel cannot move any more – without external pushing – at thisslip ratio. In this case, the optimum effective driving force Tdopt is calculated as 0.097 kNat an optimum slip ratio idopt = 10%. The analytical simulation results agree well with theexperimental test data, as shown in this diagram indicated by the ◦ and • plotted values.

Figure 2.21 shows the experimentally derived relationship between the total amount ofsinkage s and the slip ratio id . The sinkage s increases gradually with increments of id dueto increasing amount of slip sinkage till id ∼= 55%. After this the sinkage increases rapidly.The analytical results agree well with the experimental test result in this case as shown bythe computed value • marks on the figure.

Figure 2.22 shows the relationship between the amount of eccentricity ed of verticalreaction force and the slip ratio id . The parameter ed increases parabolically with increasingid , after taking a minimum value edmin = 4.2 cm at id ∼= 0%, due to the increasing amountof slip sinkage.

Figure 2.23 shows the relationships between the entry angle θf , the exit angle θr and theslip ratio id . The parameter θf increases parabolically with increments of id after takinga minimum value θfmin = 0.629 rad at id ∼= 0%. In contrast, the value of θr increases graduallywith id . It rises from θr = 0.175 rad at id ∼= 0% and decreases rapidly after reaching amaximum value θrmax = 0.244 rad.

Figure 2.20. Relationships between driving force Qd /R, effective driving force Td and slip ratio id .

72 Terramechanics

Figure 2.21. Relationship between total amount of sinkage S and slip ratio id during driving action.

Figure 2.22. Relationship between amount of eccentricity ed and slip ratio id during driving action.

Figure 2.24 shows the relationships between the various energy values E1, E2, E3, E4 andthe slip ratio id . The effective input energy E1 increases gradually with increments in id andincreases rapidly from id ∼= 55%. The sinkage deformation energy E2 increases parabol-ically with id from an initial value of 0.934 kNcm/s at id ∼= 0%. The slippage energy E3

increases almost linearly with id but increases rapidly from id ∼= 55%. The effective draw-bar pull energy E4 increases with id and reaches a maximum value E4max = 2.359 kNcm/sat id = 10%. After that, E4 decreases almost parabolically till it develops negative values.

Figure 2.25 shows the relationship between the tractive power efficiency Ed and the slipratio id . Ed decreases almost hyperbolically with id from a maximum value Edmax = 68.1%at id ∼= 0% and takes on negative values from id = 43%. Figure 2.26 shows the distributionsof normal stress σ(θ) and shear resistance τ(θ) at idopt = 10%. The shape of these stress


Figure 2.23. Relationships between entry angle θf , exit angle θr and slip ratio id during driving action.

Figure 2.24. Relationship between energy values E1, E2, E3, E4, and slip ratio id during driving action.

distribution agrees well with the experimental test data presented in a paper published byOnafeko et al. [22]. In this case, a maximum value of normal stress σmax = 156.2 kPa isobtained at θN = 0.300 rad and θN /θf = 0.450.

Figure 2.27 shows the relationship between the angle ratio θN /θf to obtain the maximumnormal stress and the slip ratio id . The ratio θN /θf increases slightly with increasing valuesof id , and the constants in Eq. (2.17), a = 0.391 and b = 1.82 × 10−3 are obtained. It isclear from this graph that the position showing the maximum normal stress shifts forward

74 Terramechanics

Figure 2.25. Relationship between tractive efficiency Ed and slip ratio id .

Figure 2.26. Distributions of contract pressure; normal stress σ and shear resistance τ, at the optimumslip ratio idopt = 10%.

39.1% of the entry angle and that it increases slightly with increasing slip ratio. Hiromaet al. [23] also observed the same experimental results.

2.4.2 Braking state

In a similar manner to the foregoing and as a further example, the braking performanceof a towed rigid wheel with axial load W = 1.52 kN, radius R = 16 cm, width B = 9.5 cmrunning on a soft sandy soil ground at a wheel speed V = 7.07 cm/s has been simulated.Likewise the mathematical simulation results have been verified by comparison with exper-imental test data. All the terrain-wheel system constants and the soil properties are the sameas given in the previous session 2.4.1 example for the driving state.


Figure 2.27. Relationship between angle ratio θN /θf and slip ratio id during driving action.

Figure 2.28. Relationship between braking force Qb/R, effective braking force Tb and skid ib duringbraking action.

The experimental towing test of the rigid wheel was executed under conditions of planestrain and the braking torque Qb, the effective braking force Tb and the total amount ofsinkage s were measured directly.

Figure 2.28 shows the relationship between the braking force Qb/R, the effective brak-ing force Tb and the skid parameter ib. Qb/R decreases rapidly with increments of |ib|and takes a zero value at ib ∼= −9%. Afterwards, |Qb/R| increases parabolically with |ib|.It reaches a maximum value |Qb/R|max = 0.674 kN at ib = −52%. |Tb| also decreases rapidlywith increasing values of |ib| and it reaches zero at ib = −7%. Afterwards, |Tb| increasesparabolically with |ib|. It increases rapidly at skid values past |ib| ∼= 70% due to increasingland locomotion resistance.

76 Terramechanics

Figure 2.29. Relationship between total amount of sinkage S and skid ib during braking action.

Figure 2.30. Relationships between amount of eccentricity eb of vertical reaction and skid ib duringbraking action.

In this situation, the optimum effective braking force Tbopt is calculated as −0.737 kN at anoptimum slip ratio ibopt = −27%. The analytical simulation results compare well with theexperimental test data, as illustrated by the ◦ and • calculated values.

Figure 2.29 shows the relationship between the total amount of sinkage s and skid ib.Initially, the sinkage s is almost constant with increasing |ib| due to the balance of the


Figure 2.31. Relationships between entry angle θf , exit angle θr and skid ib during braking action.

decreasing amount of static sinkage and the increasing amount of slip sinkage. Howeverthe sinkage begins to increase rapidly with |ib| at skid values greater than |ib| ∼= 70% due toincreasing amounts of slip sinkage. The analytical results agree well with the experimentaltest results as shown by the • marks in the diagram.

Figure 2.30 shows the relationship between the amount of eccentricity eb of the verticalreaction and skid ib. The value of eb initially decreases gradually with increments of |ib| aftertaking the maximum value ebmax = 5.3 cm at ib = −3% due to the decreasing amount of staticsinkage. After this it takes a minimum value ebmin = 3.5 cm at ib = −33%. Afterwards, eb

increases gradually with |ib| until it reaches an upper peak value eb = 4.7 cm at ib = −69%.Figure 2.31 shows the relationship between the entry angle θf , the exit angle θr and skid

ib. The value of θf increases parabolically with increasing |ib| after taking a minimum valueθfmin = 0.630 rad at ib ∼= −1%. On the other hand, θr decreases gradually with increasingvalues of |ib| from a maximum value θrmax = 0.189 rad at ib ∼= −3%.

Figure 2.32 shows the relationships between the several energy components E1, E2,E3, E4 and the skid parameter ib. The effective input energy E1 decreases rapidly withincrements in |ib| and reaches zero at ib = −9%. Afterwards, |E1| increases parabolicallywith |ib| and reaches a maximum value |E1|max = 3.197 kNcm/s at |ib| = 27%. After that,|E1| decreases gradually to zero. In contrast, the sinkage deformation energy E2 increasesparabolically with |ib|, after reaching a minimum value E2min = 0.810 kNcm/s at ib ∼= −33%.The slippage energy E3 increases almost linearly with increments of |ib| past |ib| ∼= 9%. Theeffective braking force energy E4 decreases rapidly with increases in |ib| and reaches zero atib ∼= −7%. Afterwards, |E4| increases rapidly with |ib|. It increases suddenly at skid valuesof more than |ib| ∼= 70%.

Computationally, the optimum braking force (Qb/R)opt = −0.619 kN, the optimum effec-tive braking force Tbopt = −0.737 kN, the total amount of sinkage s = 3.1 cm, the amount ofeccentricity of the vertical reaction eb = 3.6 cm, the braking power efficiency Eb = 163%,

78 Terramechanics

Figure 2.32. Relationships among energy values E1, E2, E3, E4 and skid ib during braking action.

the entry angle θf = 0.635 rad and the exit angle θr = 0.158 rad can be obtained for theoptimum skid ibopt = −27% that might be used to maximize the effective input energy E1.

Figure 2.33 shows the distributions of normal stress σ(θ) and shear resistance τ(θ) atvalues of ib = −10, −20 and −30%. The shape of these stress distribution agrees wellwith experimental results published by Onafeko et al. [22], Krick [24] and Ogaki [25]. Thedistribution of the shear resistance τ(θ) at ib = −10% shows a change from a positive valueto a negative one at a central angle θ = 0.271 rad. In contrast the distributions of τ(θ) atib = −20% and −30% are always negative.

In these cases, the maximum values of normal stress σmax = 173.6, 166.5 and 165.4 kPaare obtained at central angles θN = 0.277, 0.238 and 0.096 rad and θN /θf = 0.425, 0.375and 0.150, respectively.

Figure 2.34 shows the relationship between the angle ratio θN /θf to obtain maximumnormal stress and skid ib. The ratio θN /θf decreases gradually with increasing valuesof |ib|. The ratio takes a minimum value (θN /θf )min = −0.209 at ib = −40% and thenincreases slightly with |ib|. This tendency can be compared to the experimental test datapresented by Oida et al. [26]. In this case, it is clearly shown that the position showingthe maximum normal stress shifts forward 24.0% of the entry angle at an optimum slipvalue ibopt = −27%.

2.5 SUMMARY

In this chapter, we have studied the simplest of the machine-terrain interaction problemsnamely that of predicting the behaviour of a, loaded, rigid cylindrical drum operating upona compressible, and potentially yielding, medium. The prediction is made by developingmathematical models of the system.


Figure 2.33. Distributions of contact pressure; normal stress σ and shear resistance τ at (a) ib = −10%,(b) ib = −20% and (c) ib = −30% during braking action.

80 Terramechanics

Figure 2.34. Relationship between angle ratio θN /θf and skid ib during braking action.

Up to now a number of different simple drum/terrain systems have been modelled. Theseinclude:– A static, loaded, cylindrical drum operating on an elasto-plastic and/or visco-plastic

medium.– Heavy cylindrical drums that are towed over hard, visco-plastic and/or sandy surfaces.– Heavy, self-powered, cylindrical drums that drive themselves over hard, visco-plastic

and/or sandy surfaces.– Heavy, moving, cylindrical drums that brake themselves over hard, visco-plastic and/or

sandy surfaces.

The cylindrical drum used in the modelling process can be thought of as the drum of anearthworks compactor or of an old fashioned steam-roller. Alternately it may be the steeltire or rigid wheel of a truck or trailer. While the use of rigid wheels may seem somewhatartificial to many, the assumption of rigidity makes the problem more mathematicallytractable and gives experimentally testable results. The analyses presented here are generallytwo-dimensional only and the end effects of the drum and other three-dimensional effectshave been typically ignored in the interests of mathematical tractability.

The chapter also continues the ideas developed in Chapter 1, namely that of the use ofmetrics to characterise a terrain and a particular load state. In this chapter, the ideas havebeen systematised by use of a matrix of terrain-wheel system constants.

REFERENCES

1. Al-Hussaini, M.M. & Gilbert, P.A. (1975). On the Stress Distribution Beneath a Circular RigidWheel. Proc. 5th Int. Conf. ISTVS, Vol. 2, Michigan, U.S.A., pp. 335–365.

2. Ito, N. (1975). Theoretical Analysis of the Forces Acting about a Wheel Including Slip Sinkage.Proc. 5th Int. Conf. ISTVS, Vol. 2, Michigan, U.S.A., pp. 311–333.

3. Wong, J.Y. & Reece, A.R. (1966). Soil Failure beneath Rigid Wheels. Proc. 2nd Int. Conf., ISTVS,Quebec, Canada, pp. 425–445.

4. Yong, R.N. & Windish, E.J. (1970). Determination of Wheel Contact Stresses for MeasuredInstantaneous Soil Deformation. J. of Terramechanics, Vol. 7, No. 3/4, pp. 57–67.

5. Yong, R.N. & Fattah, E.A. (1976). Prediction of Wheel Soil Interaction and Performance usingthe Finite Element Method. J. of Terramechanics, 13, 4, pp. 227–240.


6. Harrison, W.L. (1975). Shallow Snow Performance of Wheeled Vehicles. Proc. 5th Int. Conf.ISTVS, Vol. 2, Michigan, U.S.A., pp. 589–614.

7. Akai, K. (1986). Soil Mechanics. pp. 169–192, Asakura Press, (In Japanese).8. Terzaghi, K. (1943). Theoretical Soil Mechanics. pp. 118–143. John Wiley & Sons.9. Terzaghi, K. (1943). Theoretical Soil Mechanics. pp. 367–415. John Wiley & Sons.

10. Janosi, Z. & Hanamoto, B. (1961). The Analytical Determination of Drawbar Pull as a Functionof Slip for Tracked Vehicle. Proc.1st Int. Conf. on Terrain-Vehicle Systems, Torio.

11. Bekker, M.G. (1961). Off-the-Road Locomotion. pp. 58–66. The University of Michigan Press.12. Kacigin, V.V. & Guskov, V.V. (1968). The Basis of Tractor Performance Theory. J. of Terra-

mechanics, 5, 3, pp. 43–66.13. Terzaghi, K. & Peck, R.B. (1948). Soil Mechanics in Engineering Practice. pp. 167–175. John

Wiley & Sons.14. Wong, J.Y. & Reece, A.R. (1967). Prediction of rigid wheel performance based on the analysis

of soil-wheel stresses – Part I. Performance of driven rigid wheels. J. of Terramechanics, 4, 1,pp. 81–98.

15. Wong, J.Y. (1967). Behaviour of soil beneath rigid wheels. J. of Agric. Engng. Res., 12, 4,pp. 257–269.

16. Yong, R.N. & Fattah, E.A. (1976). Prediction of wheel-soil interaction and performance usingthe finite element method. J. of Terramechanics, 13, 4, pp. 227–240.

17. Yong, R.N. & Fattah, E.A. (1975). Influence of Contact Characteristics on Energy Transfer andWheel Performance on Soft Soil. Proc. 5th Int. Conf. ISTVS, Vol. 2, Detroit, U.S.A., pp. 291–310.

18. Mogami, T. (1969). Soil Mechanics. pp. 479–622, Gihoudou Press, (In Japanese).19. Wong, J.Y. & Reece, A.R. (1967). Prediction of Rigid Wheel Performance Based on the Analysis

of Soil-Wheel Stresses – Part II. Performance of Towed Rigid Wheels. J. of Terramechanics, 4, 2,pp. 7–25.

20. Scott, R.F. (1965). Principles of Soil Mechanics. pp. 398–472. Addison-Wesley PublishingCompany.

21. Wong, J.Y. (1978). Theory of Ground Vehicles. pp. 55–121. John Wiley & Sons.22. Onafeko, O. & Reece, A.R. (1967). Soil Stresses and Deformations beneath Rigid Wheels.

J. of Terramechanics, 4, 1, pp. 59–80.23. Hiroma, T. & Ohta, Y. (1987). A Measurement of Normal and Tangential Stress Distribution

under Rigid Wheel. Terramechanics, Vol. 7, pp. 7–13. The Japanese Society for Terramechanics,(In Japanese).

24. Krick, G. (1969). Radial and Shear Stress Distribution under Rigid Wheels and Pneumatic TiresOperating on Yielding Soils with Consideration of Tire Deformation. J. of Terramechanics, 6, 3,pp. 73–98.

25. Ogaki, M. (1984). The Normal and Tangential Stress Distribution Acting on the Contact Surfaceof the Rigid Wheel. Terramechanics, Vol. 4, pp. 12–15. The Japanese Society for Terramechanics,(In Japanese).

26. Oida, A., Satoh, A., Ito, H. &Triratanasirichai, K. (1991). Three Dimensional Stress Distributionson a Tire–Sand Contact Surface. J. of Terramechanics, 28, 4, pp. 319–330.

EXERCISES

(1) A rigid wheel of radius of R = 10 cm is running in straight forward motion underdriving action on a sandy terrain at a rotational speed Rω = 30 cm/s and with a slipratio of id = 10%. Calculate the total rolling time t and the total moving loss j duringthe rolling distance of 10 m.

(2) A rigid wheel of radius of R = 20 cm is running during driving action on a soft groundat a rotational speed Rω = 50 cm/s and with a slip ratio of id = 20%. The contact lengthof the rigid wheel against the terrain is measured as 10 cm. Calculate the transit time

82 Terramechanics

te of the wheel over an arbitrary point X of the ground and the amount of slippage jsof the soil at the point X .

(3) A rigid wheel is running during driving action on a soft ground at a slip ratio ofid = 15%. The radius of the wheel R is 8 cm, the speed of rotation Rω is 40 cm/s andthe entry angle θf is π/6 rad. Calculate the slip velocity Vs and the amount of slippagejd at the bottom-dead-center of the wheel.

(4) A rigid wheel is running during driving action on a soft ground at a slip ratio ofid = 30%. The entry angle θf at the beginning of contact of the wheel to the groundis π/6 rad. Can you show that the direction of the resultant velocity vector of a soilparticle located on the contact point A is coincident with the gradient of tangent to therolling locus e.g. trochoid curve of the wheel at this point?

(5) A rigid wheel of radius of R = 50 cm is running during driving action on a soft groundat a slip ratio of id = 25%. All the soil particles on the contact part of the wheel arealways rotating around the instantaneous center I. Calculate the coordinates of thecenter I relative to the origin of the wheel axle O.

(6) A rigid wheel of radius of R = 20 cm is running straight forward during braking actionon a sandy terrain at a rotational speed of Rω = 50 cm/s and at a skid of ib = −20%.Calculate the total rolling time and the total moving loss during the rolling distanceof 10 m.

(7) A rigid wheel of radius of R = 10 cm is running during braking action on a soft groundat a rotational speed of Rω = 30 cm/s and with skid of ib = −15%. The contact lengthof the rigid wheel to the terrain is measured as 10 cm. Calculate the transit time te ofthe wheel on an arbitrary point X of the ground and the amount of slippage js of thesoil at the point X .

(8) A rigid wheel is running during braking action on a soft ground at the skid ib = −20%.The radius of the wheel R is 10 cm, the speed of rotation Rω is 50 cm/s and the entryangle θf is π/6 rad. Calculate the slip velocity Vs and the amount of slippage jb at thebottom-dead-center of the wheel.

(9) A rigid wheel is running during braking action on a soft ground at a skid of ib = −30%.The entry angle θf at the beginning of contact of the wheel to the ground is π/6 rad.Demonstrate that the direction of the resultant velocity vector of a soil particle locatedon the contact point A is coincident with the gradient of tangent to the rolling locuse.g. trochoid curve of the wheel at this point.

(10) A rigid wheel of radius of R = 30 cm is running during braking action on a soft groundat a skid of ib = −50%. All the soil particles on the contact part of the wheel are alwaysrotating around the instantaneous center I. Calculate the coordinates of the center Ifrom the origin of the wheel axle O.

Chapter 3

Flexible-Tire Wheel Systems

In 1888, the Scottish inventor John Dunlop submitted a patent for a pneumatic tire fora tricycle. Since that time, many different forms of pleasure and industrial vehicles withsoft and fluid-pressure tires have been developed. In construction and mining, especially,many common forms of excavation and loading machines such as, wheel-loaders, backhoes,tractor shovels and skid-steer loaders come equipped with special types of heavy duty rubbertires. Also, many pieces of modern civil engineering and mining haulage equipment, such asrigid and articulated-frame dump-trucks, motor-graders and rubber tired roller-compactors,operate through multiple sets of on-the-road or off-the-road rubber tires. Because these airor fluid-filled rubber tires are deliberately designed to deform with axle load or wheeltorque, they are formally referred to as ‘flexibly tired wheels’.

The interaction problem between a flexible tire and a particular terrain is an engineeringproblem that has been studied for many decades. In particular, experiments and theoryhave been developed to define the relations that exist between axle load, torque and groundreaction and the contact pressure distribution of the tire taking into consideration the com-pression and shear deformation characteristics of the terrain. Additionally, the relationshipbetween thrust or drag, land locomotion resistance, effective driving or braking force andslip ratio or skid have been extensively studied for various types of terrain materials duringboth wheel driving and/or braking action.

For hard terrains, there exists a body of research which principally concerns itself withthe motion of revolution of a tire, the kinematic straight forward motion equations and thetire’s cornering characteristics during driving and braking action. For instance, Komandi [1]studied the circumferential force of a tire acting on a concrete pavement. Muro [2] concludedthat the frictional work that develops between the tire of a heavy dump truck and a terrainwill depend on the longitudinal and lateral amount of slippage. He also studied the corneringcharacteristics of the off-the-road tire.

However, to properly analyse the trafficability problem of a tire running on a soft terrain,it is necessary to take into account variations in the shear strength and deformation propertiesof the terrain. The reason for this is that various compaction and remolding effects can occurduring passage of a tired wheel. These effects do not occur with very hard terrains. For softterrains, Yong et al. [3] developed a new analytical process, based around the finite elementmethod (FEM), which potentially can analyse the interaction problems between a tire anda clayey terrain. They consider the effect of the flexibility of the carcass of the tire on thedistribution of the contact pressure which acts on the peripheral surface of the tire. Theyalso claim that the analytical results agree well with available experimental data for variousslip ratios. For soft ground Fujimoto [4] has analysed theoretically some matters relatingto driving torque, rolling resistance and rut depth for a tire running in driving mode on aclayey soil. Forde [5] has portrayed the relationship between effective tractive effort anddriving torque on a tire by use of several contour lines that utilise the parameters of slipratio, amount of slippage and vertical ground reaction. Forde demonstrated that a higher

84 Terramechanics

effective tractive effort could be developed for a clayey soil having higher plastic indexrather than a lower plastic index even if the consistencies and the water contents of boththe clayey soils had the same values.

In the following sections of this Chapter, some methods for predicting the distribution ofcontact pressure acting on the peripheral surface of a tire as well as the fundamental charac-teristics of trafficability and the cornering characteristics of a tire will be developed basedupon the mechanical characteristics of the structure of a tire. Also some aspects of land loco-motion mechanics that utilise the equations of the movement of a tire running on a hard orsoft terrain will be discussed. Further, a mathematical simulation method will be describedthat allows calculation of the trafficability of a tire during driving and braking actions to beundertaken. The simulation method will be then used to develop several analytical results.

3.1 TIRE STRUCTURE

An off-the-road tire, of the kind typically used in the construction and mining industries,has three principal functions. The first function is to maintain trafficability of the tire whenit is sustaining high axle loads and when it is operating over soft and/or muddy terrains.

The second function is the relief of impact loads on, or generated by, the tire as they occuron uneven rough terrain such as rock masses or in forests. The third function is to guaranteethe tight propagation of the driving and braking torques and the cornering forces that occurduring turning motions on a terrain. Additionally, any vehicle carrying a large amount ofcargo needs a high payload tire. Also, any vehicle running on a soft terrain needs a widebase tire of low contact pressure and high buoyancy. Further, when an off-the-road tire isrunning over a rocky terrain – such as blasted stone – or over a stump, a high resistanceto abrasive wear or cut development is typically required of the tire. Moreover, where anoff-the-road tire is required to run at high speed for a long period, a tire having a high heatresistance is demanded.

As shown in Figure 3.1, the structure of an off-the-road tire consists of various parts:namely the crown, shoulder, side wall and bead. The main surrounding part of the tire iscalled the ‘carcass’or ‘casing’. This part is required to sustain, a typically high, air pressure.The carcass has an important role to play in the development of effective driving or brakingforce in a tired wheel system by propagating the axle load, and the driving and brakingtorques to the terrain. It is also required to pass the ground thrust-reaction, drag, landlocomotion resistance, and cornering forces through to the wheel.

Usually, a carcass is produced in several discrete stages. Firstly, rubber coated sheetsare made through a gluing process in which ‘cloths’ made of cross folded tire cords (suchas cotton, nylon or polyester) are developed and in which the voids and interstices arefilled with rubber material. Then, a number of these rubber coated sheets are stuck togetheralternately with rubber material sandwiched between them. Bias tires belong to a group oftires in which the direction of the rubber coated sheets is biased to the central line of tread.On the other hand, radial tire belongs to another group of tire wherein the rubber coatedsheets are directed radially i.e. at right angle to the central line of tread, and the carcasslayers are bound together by several belts.

On the crown part of the tire, a thick rubber tread is used to protect the carcass. Therubber materials that comprise the tread must have a high resistance to abrasive wear, to cut

Flexible-Tire Wheel Systems 85

Figure 3.1. Structure of tire.

development and to the large amounts of heat that builds-up when a large torque is appliedbetween the tire and terrain and when large amounts of slippage occur. For the varioustypes of demand situation that occur in relation to the operation of off-road tires, severaldifferent configurations of tread pattern comprised of thick rubber plates and deep grooveshave been developed.

On the outer part of the tire, a number of nylon cords or steel wires are used to reinforceand compensate for the large differences in rigidity that occur between crown and carcass.To protect the carcass, nylon cord is commonly used as a cushion material for bias tiresand likewise steel wire is inserted in a part of the belt for radial tires. On the part of carcasswhich forms the skeleton of the tire, several sheets of nylon carcass are stuck to each otherfor bias tires whilst a steel carcass is used for radial tires.

On the sidewall part of the tire, special rubber materials which have a high bending resis-tance and a high weathering resistance are used to protect the deformation of the carcass.On the bead part of the tire, several ring shaped bead wires which are made of steel are usedto tightly fix the tire carcass to the rim against the action of the high internal air pressure.

In the most common arrangement, the ply layers of the carcass are folded and woundaround the bead wire.

To characterise the size and strength of an off-the-road tire, a designation of width oftire/flatness-rim diameter-ply rating (PR) like 45/65-45-50PR is used. The measuring unitsused in the tire industry for width of tire and rim diameter are inches. The flatness ratio(i.e. the tire aspect ratio) is defined as the ratio of height of the tire to the width of tireexpressed as a percentage.

Figure 3.2 shows several representative tread patterns for various types of off-the-roadtire. Sketch (a) shows a rib type tread which has several grooves that run parallel to thelongitudinal direction of the tire. This configuration produces a high resistance against

86 Terramechanics

Figure 3.2. Tread patterns.

lateral slippage which is a property that gives a superior stability of operation to the tire.This tread pattern is often used for the front tires of motor graders because of the smallrolling resistance and good lateral stability. Sketch (b) shows a rock type tread which istypically used for the tires of heavy dump trucks or shovel loaders which may be required tooperating at quarry sites. The tread rubber in this case is of a kind that has a high resistanceto abrasive wear and to the development of cuts. Sketch (c) illustrates a traction type treadwhich is designed to tightly transfer or couple driving and braking torques to the terrain. Inthis type, the direction of the groove is inclined to the central axis of the tread. This treadconfiguration is considered to be most effective for a driven tire when it is set up in a directionsuch that it discharges the soil, while for a towed tire it is best used set up in the reversedirection. Sketch (d) illustrates a block type tread which is massed with buttons and which isdesigned to have a high floatation. As contact area increases with increasing values of axleload, the bearing capacity e.g. the high buoyancy on a soft terrain, is maintained. In contrast,sketch (e) illustrates a smooth type tread which has no grooves on the peripheral surfaceof the tire. This type of tread pattern is used for the tires of compaction machinery, such asrubber tired rollers, which are required to develop high levels of compaction of earthfill.

3.2 STATIC MECHANICAL CHARACTERISTICS

When a (off-the-road) tire of radius R stands on a steel plate while sustaining an axle loadW , as shown in Figure 3.3, the contact area A can be expressed by Eq. (3.1). This equation,however, idealises reality to a degree, in that it assumes that the tire deforms only at thecontact interface with the terrain and that the contact pressure equals the tire air pressure p.

a = √2Rf

b = √2rf

A = πab = 2πf√

Rr (3.1)


Figure 3.3. Deformation of contact part of tire on rigid plate.

Figure 3.4. Relationship between static axle load W and amount of deformation f of buff tire [6].

and hence

W = 2πf√

Rr p (3.2)

where r is the radius of the tread crown, f is the amount of deformation of the tire, anda and b are the major and minor axes of an elliptical contact area. From Eq. (3.2), it canbe seen that the sustainable axle load W for a given tire pressure, is proportional to the tiredeformation f .

To confirm or deny this relation, Figure 3.4 shows some experimental test results whichYong et al. [6] obtained for a treadless buff tire of 4.10/3.50-4.00-2PR in an study theyundertook to discover the relation that exists between W and f . Their test results confirm,to a very good degree, the above relation Eq. (3.2) for various tire air pressures p.

Unfortunately, however, this simple idealised model is not valid for actual off-the-roadtires because these are equipped with several types and thicknesses of tread rubber.

In the off-the-road tire situation, the amount of deformation of the tire f is made up of adeformation of the tread rubber, a deformation of the carcass of the tire and a compressivedeformation of the contained air. For this more complex situation, the axle load W can beexpressed as a non-linear relation, such as Eq. (3.3) which involves the total amount ofdeformation f .

W = f 2

C1 + C2{ f /( p + p0)} (3.3)

88 Terramechanics

In this equation, C1 is a constant which depends on the radius of the tire R, on the radiusof the tread crown r, on the amount of deformation of the tire f , on the thickness of thetread, on the elastic modulus of the tread rubber, on the coefficient of fillness of the treadarea in relation to the total contact area, on the coefficient of contact pressure distributionand on the stiffness of the tread rubber against lateral deformation. The parameter C2 is acoefficient which can be determined from data relating to the radius of the tire R, to theradius of the tread crown r, and to the ratio of the volume of the elliptical segment at thecontact part to the inside volume of the tire. The factor p is the tire air pressure and p0

is defined as the ratio of the deformation work of the carcass at zero air pressure to theamount of variation of the inside volume of the tire.

Yong et al. [7] studied analytically the effects of elastic modulus and shear modulusof rigidity of a tire assuming a two-dimensional situation where two elastic cylinders arerolling. The cylinders have a contact length of 2a and sustain an axle load of P. The resultsof their study are contained in the following set of expressions.

1

Rr+ 1

Rs= 4P

πa2

(1 − ν2

s

ET+ 1 − ν2

T

Es

)/(1 + k2

s

k2T

)

kT = 2

π

(1 − νT

GT+ 1 − νs

Gs

)

ks = 1 − 2νs

GT− 1 − 2νs

Gs

GT = ET

2(1 + νT )

Gs = Es

2(1 + νs)(3.4)

For the particulars of this analytical case, RT is the radius of the tire and Rs has a valueof ∞. ET is the elastic modulus of the tire, Es is the elastic modulus of the terrain, GT isthe shear modulus of rigidity of the tire, Gs is the shear modulus of rigidity of the terrainand νT and νS are the Poisson’s ratio of the tire and the terrain respectively. The value ofthe parameter νt can be taken to have a value of 0.5 for the negligibly small variation inthe volume of the tire that applies in this situation. The values of ET and GT of the tire canbe calculated subsequent to measuring the values of Es, νs and a for a particular terrain.

Abeels et al. [8] developed an experimental apparatus as shown in Figure 3.5 to investigatethe mechanical properties of tires. When a test tire was placed in a soil bin, which was filledwith various kinds of soil material, and a test axle load W and torque Q was applied, theresulting contact length 2a, the height of tire Hw and the width of tire Bw could be observed.Using these observations the researchers calculated a ratio of compression th and a ratio offlatness tB of the tire using the following equations:

th = H0 − Hw

H0× 100 (%) (3.5)

tB = B0 − Bw

B0× 100 (%) (3.6)

Here, H0 and B0 are the initial height and the initial width of the tire, respectively.


Figure 3.5. Measuring apparatus for elastic modulus of tire [8].

Figure 3.6. Test apparatus for deformation of tread and lateral rigidity of tire [8].

The same research group also developed another experimental apparatus – as shown inFigure 3.6. Through the use of this set-up they were able to investigate the lateral rigidityof tires and the characteristics of deformation of the tire’s tread due to lateral load.

In this case, the lateral shear modulus of rigidity of a tire can be obtained from the shearstrain calculated for the measured amount of deformation of the loaded point to which acornering force T is applied.

The elastic modulus of the tread can be obtained by measuring the normal amount ofdeformation of the tread when a normally directed load is applied. In this specific pieceof apparatus, the normal force is generated by use of a hydraulic jack via a lever which ispivoted on the upper frame. Under these same circumstances, the shear modulus of rigidityof the tread can also be obtained by measuring the amount of deformation of the loadedpoint when both a normal and a tangential force are applied simultaneously. These twinforces are generated by use of two hydraulic jacks via two levers pivoted on the upper andside frame respectively.

Yong et al. [7] determined that the elastic modulus ET of tires equipped with several typesof grooved tread rubber increases linearly with increasing tire air pressure p for constantaxle load. Typical data to illustrate these results is given in Figure 3.7.

90 Terramechanics

Figure 3.7. Relationship between modulus of elasticity ET of tire and air pressure p [7] (tread tire4.10/3.50-4.00-2pk).

Figure 3.8. Relationship between deformation energy EdT of tire and relative rigidity of terrain totire Es/ET .

Yong et al. also used the finite element method, to demonstrate that the deformation energyEdT of a tire varies with the ratio of the elastic modulus of the terrain Es to the elasticmodulus of the tire ET for a constant axle load and tire air pressure [6]. Their results areshown in Figure 3.8. In this case, Es/ET = 1.0 can be considered to be a boundary in judgingwhether a tire is rigid or flexible. Thus, a tire may be deemed to act as a rigid wheel whenEs is less than ET .

Fujimoto [9] determined that a critical tire air pressure p could be developed as thedifference between the average contact pressure q under an axle load W and the rigidcontact pressure of the carcass pc which exists when a tire is self-standing by virtue of itsown rigidity and when there is no applied axle load. The critical tire air pressure is the tireair pressure at which a tire changes from a rigid tire into a flexible tire.

A tire can be considered to be a rigid tire when the relation of Eq. (3.7) applies:

p ≥ q − pc (3.7)


Figure 3.9. Ground reaction during pure rolling state of flexible tired wheel on hard terrain.

In these studies, it is very important to standardize test methods for the static mechanicalcharacteristics of tires such as the elastic modulus and the shear modulus of rigidity. It isespecially necessary to clearly define the mutual relationships between the tire air pressure,the elastic modulus and the shear modulus of rigidity of the tire.

3.3 DYNAMIC MECHANICAL PROPERTIES

3.3.1 Hard terrain

When a flexible tired wheel is running upon a hard terrain – such as an asphalt or concretepavement, the amounts of deformation of the tread rubber and the carcass of the tire becomevery large, while the amounts of deformation in the terrain are negligibly small.

Suppose that we now analyse the force balances that apply when a tire is running on ahard terrain at a constant speed V with a constant rotation speed Rω and that we undertakethis analysis alternately for the pure rolling state and for the driving and braking states.

In this situation we can note that for the pure rolling state, some compressive deformationwill occur symmetrically on both sides of the contact part of the tire due to the axle loadW . Also, as shown in Figure 3.9, the amount of slippage j at the end of the contact part of atire of length L during braking action can be given as ibL/(1 + ib) for a skid value ib (<0).As well, the distributions of contact pressures e.g. the normal stress σ and shear resistanceτ, deviate a little from their rest positions in the moving direction of the tire.

In the externally propelled rolling wheel case, the vertical ground reaction N must equalthe axle load W and the effective braking force Tr acting in the moving direction of the wheelmust equal the horizontal ground reaction µrW i.e. the rolling resistance. The direction ofthe resultant force comprised of N and µrW goes through the central axis of the wheel. Theposition of application of the resultant force can be expressed in terms of a vertical effective

92 Terramechanics

Figure 3.10. Ground reaction during driving action of flexible tired wheel on hard terrain.

rolling radius Re0 and a horizontal amount of eccentricity er . Using these parameters, themoment balance around the axle can be expressed as:

Ner = µrWRe0 (3.8)

and hence

er = µrRe0 (3.9)

where µr is the coefficient of rolling friction.For a flexibly tired wheel operating in a driving state, the forward part of the carcass

to the moving direction will expand due to compression effects whilst the rear part of thecarcass will be extended due to tension effects upon the application of an axle load W anda driving torque Qd . As shown in Figure 3.10, the amount of slippage j at the end of thecontact part of the tire of the length L during driving action can be given as idL for a slipratio id (>0). In this case, the distributions of contact pressures e.g. normal stress σ andshear resistance τ deviate towards the moving direction of the tire. From considerations offorce equilibrium, the vertical ground reaction N must equal the axle load W whilst theeffective driving force Td must equal the horizontal ground reaction (µd − µr)W – whereµd is the coefficient of driving resistance.

The position of application of the resultant ground reaction in this case can be expressedin terms of a vertical effective rolling radius Red and a horizontal amount of eccentricityed . Thence, from moment equilibrium around the axle we have:

Qd = µdWRed

Qd − Ned − (µd − µr)WRed = 0 (3.10)

and hence

ed = µrRed (3.11)


Figure 3.11. Ground reaction during braking action of flexible tired wheel on hard terrain.

In the initial stage of application of a small driving torque Qd , µd < µr the direction ofaction of the effective driving force Td is coincident with the moving direction of the wheel.When the driving torque Qd increases more and µd = µr , Td becomes zero and the wheelwill run in a self-propelling state. For a large driving torque Qd , the effective driving forceTd applies oppositely to the moving direction of the wheel as shown in the diagram.

Harlow et al. [10] determined that all the shear resistance vectors acting on the treadsurface were located in the moving direction of the wheel and along the longitudinal centerline of the contact part of the tire.

For a flexibly tired wheel operating in braking mode, the forward part of the carcass tothe moving direction will be extended due to tension effects and the rear part of the carcasswill be expanded due to compression effects through the application of an axle load W anda braking torque Qb. As shown in Figure 3.11, the amount of slippage j at the end of thecontact part of the tire of length L during braking action may be given as ibL/(1 + ib) fora skid value ib(<0). In this situation the distributions of contact pressures e.g. the normalstress σ and shear resistance τ progress towards the moving direction of the tire. Also, thevertical ground reaction N equals the axle load W and the effective braking force Tb equalsthe horizontal ground reaction (µb+µr)W where µb is the coefficient of braking resistance.In a similar manner to the above cases, the position of application of the resultant groundreaction can be expressed in terms of a vertical effective rolling radius Reb and a horizontalamount of eccentricity eb. Thence, moments around the axle can be equated to yield thefollowing equation.

Qb = µbWReb

−Qb − Neb + (µb + µr)WReb = 0 (3.12)

94 Terramechanics

Figure 3.12. Measured relationship between longitudinal component of friction µd and slip ratio id

of heavy dump truck [11].

therefore

eb = µrReb (3.13)

The relationships that exist between the coefficients of driving and braking resistance µd ,µb, and the slip ratio id and the skid ib for any particular off-the-road tire depend on theshape of the groove, the material and the surface roughness of the tread rubber. For example,Figure 3.12 shows the µ–i curve for an off-the-road tire for a heavy dump-truck. The datawas measured in-situ while the tire was running on a hard terrain of decomposed weatheredgranite based sandy soil [11]. The tire was of a 1.00-35-36PR type.

3.3.2 Soft terrain

Relative to the problem of determining the trafficability of an off-the-road tire that isrunning on a soft terrain, Turnage [12] proposed that the coefficient of traction and thetractive efficiency of a tire could be estimated experimentally from the next wheel mobilitynumbers Ns and Nc as developed in the following equations:

For sandy terrain:

Ns = G(BD)32

W· δ

H(3.14)

For clayey terrain:

Nc = CBD

W

(δ

H

) 12

· 1

1 + B/2D(3.15)

Here, C is the cone index of the terrain, G is the gradient of the cone index e.g. the coneindex divided by the depth of penetration, W is the axle load of the tire, B is the width ofthe wheel, D is the diameter of the wheel, H is the height of the tire, and δ is the amountof tire deformation.

Figure 3.13(a) shows the relationship between the coefficient of traction µ20, the tractiveefficiency η20 at slip ratio id = 20% and the wheel mobility number Ns for a sandy terrain.

Likewise, Figure 3.13(b) shows the relationship between µ20, η20 at id = 20% and wheelmobility number Nc for a clayey terrain.


Figure 3.13a. Relationship between coefficient of traction µ20, tractive efficiency η20 and wheelmobility number Ns [12] for a sandy terrain.

Figure 3.13b. Relationship between coefficient of traction µ20, tractive efficiency η20 and wheelmobility number, Nc [12] for a clayey terrain.

Yong et al. [3] investigated the effects of the tread pattern of a tire on the effective drivingforce during driving action for various levels of tire air pressure. As a result, it was deter-mined that a traction type tread can improve, quite remarkably, the effective driving force,whilst a more rigid tire having high air pressure can develop the most effective drivingforce for a sandy terrain. For a clayey terrain a more flexible tire having low air pressurecan develop the most effective driving force. For example, Figure 3.14 shows a relation-ship that prevails for various values of energy Edt and slip ratio id for an off-the-road tirewith mounted tread rubber. The tire is inflated to an air pressure of 40 kPa such as wouldbe indicated for a sandy terrain. From this diagram, it is evident that the slippage energy

96 Terramechanics

Figure 3.14. Relationship between energy Edt and slip ratio id of tread tire [3].

between the tire and the terrain increases parabolically with increasing values of id and thatthe output energy takes a maximum value at some value of slip ratio.

In general, the shape of the distribution of contact pressure of a tire running on a softterrain is essentially identical to that of a rigid wheel moving in pure rolling, driving orbraking states as has been already discussed in the previous Chapter. Further, Krick [13] andOida et al. [14] confirmed that contact pressure tends to decrease relatively with increasingdeformation of the contact part of the tire.

Söhne [15] investigated the effect of the stress distribution under a tire on the deformationof the terrain and on the amount of compaction of the substrate soil. In these studies, itis necessary to vary the type of tire, the axle load, the thickness of lift and the number ofcompaction passes required to achieve an optimum compaction of soil as a consequenceof the existence of a finite sized pressure bulb that occurs in the terrain under the tire. Thesize of this bulb varies with axle load and soil properties.

In these studies, it was found that a most effective method of compaction for increasingthe dry density of a soil and its shear strength could be developed by creating an alternatingshear stress in the soil. This effect could be practically obtained by developing a coupledland locomotion system comprised of a pure rolling front tire and a driven rear tire [16].

To analyse the specific mechanisms of soil compaction, Bolling [17] calculated thetrochoidal rolling locus of a tire running at various slip ratios and thence measured thevertical distribution of the normal earth pressure from a plate shear test by use of a platemounted with a given tread rubber which moved in the soil along the rolling locus.

(1) During driving action

When the tire air pressure p of a tire running on a soft terrain is larger than the differencebetween the average contact pressure q and the rigid contact pressure of the carcass pC

as described in Eq. (3.7), the tire can be treated as a ‘rigid wheel’ because the tire does


Figure 3.15. Relationship between tire air pressure p and average contact pressure q (pC = rigidcontact pressure).

Figure 3.16. Relationship between tire air pressure p and average contact pressure for various axleloads W on tire [19].

not deform. Bekker [18] proposed that the critical average contact pressure qcr whichdistinguishes between a ‘rigid wheel’ and a ‘flexible wheel’ equals the sum of the tire airpressure p and the rigid contact pressure of the carcass pC on a hard terrain, as shown inFigure 3.15. Unfortunately though and in general, the rigid contact pressure of the carcasspC on a soft terrain can not easily be determined since pC depends not only on the tireair pressure and the weight of tire but also on the properties of the soil that comprises thesoft terrain. The previous theory, as mentioned in Section 2.2, can be applied to provide anestimate for a rigid tire since the average contact pressure q, e.g. the axle load W dividedby the contact area, is less than the critical average contact pressure qcr .

To investigate these findings, Schwanghart [19] measured the contact area of an off-the-road tire 13.6/12-28 of diameter 1310 mm and of width 345 mm under various axleloads W when the tire system was operating on a loose accumulated sandy loam terrain ofwater content 15%. From these observations he developed a relationship between averagecontact pressure q and the tire air pressure p as shown in Figure 3.16. From this diagram,the tire can be seen to function as a rigid wheel in those regions where q � p + pC . Forconstant terrain conditions and for a constant axle load W , the average contact pressureq increases with increments in tire air pressure p due to corresponding decrements incontact area.

Further, for constant terrain conditions and for a constant tire air pressure p, the averagecontact pressure q tends to increase with increments in axle load W , while the contact areaalso increases with increments in W .

98 Terramechanics

Figure 3.17. Flexible deformation of tire (straight logarithmic spiral line).

When the tire air pressure p has a comparatively low value and is less than the differencebetween the average contact pressure q and the rigid contact pressure of the carcass pC , thetire can be treated as a ‘flexible wheel’ because the tire can deform.

There are many hypotheses about the shape of the flexible deformation of the contact partof the tire. Karafiath et al. [20] proposed that the shape of the deformation of the contact partof a tire could be considered to be comprised of a central straight part with logarithmic spiralline portions on both sides as shown in Figure 3.17. The radius r of the forward logarithmicspiral line can then be expressed as a function of an arbitrary central angle α, as follows:

r = R exp{β(α − α0)} (3.16)

where R is the radius of tire, β is a constant value, α0 is the central angle at the beginningof the logarithmic spiral line, and α0 = θf + π/36 rad for the entry angle θf .

The angle α is taken to lie arbitrarily in the range αe �α �α0 where αe is the centralangle at the beginning of the straight line. Using the same approach, the radius of the rearlogarithmic spiral line can be likewise calculated.

Blaszkiewicz [21] developed a real-time apparatus for measuring the depth of the rut gen-erated and the radial, longitudinal and lateral amounts of deformation of a tire running ona soft terrain. Using an assumed radial deformation function taken as a polynomial expres-sion, he made a mathematical model which expressed the total shape of the deformation ofthe contact part of the tire provided that the shape of cross section of the contact part of thetire due to the longitudinal and lateral deformation could be approximated as an ellipse.

Wong [22] proposed a somewhat simpler deformation shape for a tire using circular arcsegments as shown in Figure 3.18.

Relative to these conditions, let us now consider the overall equilibrium and force balancethat exists between the axle load W , the driving torque Qd , the effective driving force Td

during driving action, and the distributions of contact pressure α(θ), τ(θ) which act on thecontact part of a flexible tired wheel. As shown in the diagram, the central angle of thestraight part AB which is flattened due to the deformation of the tire is 2θC and the centralangles of the circular arcs BC and AD are θf − θC and θr − θC respectively. The original


Figure 3.18. Distributions of amount of slippage jd (θ) and ground reaction a(θ), τ(θ) of contact partof tire (during driving action).

radius of the tire is designated as R and the central angle of the radius vector, measuredcounterclockwise from the vertical axis, is designated as θ.

The respective amounts of slippage j = jd(θ) of the circular part BC, the straight part ABand the circular part AD of the contact part of the tire may be expressed, using the previousEq. (2.16), as ;

For θC � θ � θf

jd(θ) = R{(θf − θ) − (1 − id)(sin θf − sin θ)} (3.17)

For −θC � θ < θC

jd(θ) = jd(θC) + R(sin θC − sin θ)id (3.18)

For −θr � θ < −θC

jd(θ) = jd(−θC) + R∫ −θc

θ

{1 − (1 − id) cos θ}dθ

= jd(−θC) − R{(θC + θ) − (1 − id)(sin θC + sin θ)} (3.19)

The distributions of contact pressure σ(θ) and τ(θ) take positive values for the whole rangeof the contact area of the tire. The angle δ(θ) between the resultant pressure p(θ) and theradial direction of the tire may be defined as δ(θ) = tan−1{τ(θ)/σ(θ)}. The amount of

100 Terramechanics

sinkage s0 of the bottom-dead-center just under the central axis of the tire and the amountof rebound u0 just after the passing of the tire can be expressed as:

s0 = R( cos θC − cos θf ) (3.20)

u0 = R( cos θC − cos θr) (3.21)

Further, the resultant pressure p(θ) can be expressed, by use of an elemental length ofrolling locus d(θ) in the direction of the applied stress q(θ) as shown in previous Eq. (2.36),a coefficient of modification ξ to allow for dynamic effects, and an angle of inclination ofthe effective driving force ζ = tan−1(Td /W cos β), as follows:

For θmax � θ � θf

p(θ) = k1ξ{d(θ)}n1

cos(ς + θ − δ)(3.22)

For θC � θ < θmax


cos(ς + θ − δ)(3.23)



cos(ξ + θ − δ)(3.24)


p(θ) = ||k1ξ{d(θmax)}n1 − k2{d(θmax) − d(θC)}n2 − k2[d(θmax) − {d(θC) + d ′(θ)}]n2 ||cos(ς + θ − δ)

(3.25)

Here, θmax is the central angle θ at which p(θ) takes a maximum value and d ′(θ) is a functionwhich integrates the previous Eq. (2.36) from θ = θ to θ = −θc. In the above equations,the coefficients k1, k2 and the indices n1, n2 are a set of terrain-tire system constants whichcan be determined from standard plate loading and unloading tests.

Thence, the distribution of normal stress σ(θ) may be given by:

σ(θ) = p(θ) cos{δ(θ)} (3.26)

For −θC � θ � θC , σ(θ) can be written as follows – assuming that the applied resultant stressis constant e.g. p(θ) = pg .

σ(θ) = pg cos{δ(θ)} (3.27)

An expression for the distribution of shear resistance τ(θ) can be obtained by substitutingthe amount of slippage jd(θ) – given in the previous Eqs. (3.17), (3.18) and (3.19) – intothe previous Eqs. (2.8), (2.9). This action yields the following equation.

τ(θ) = {ca + σ(θ) tan φ}[1 − exp{−ajd(θ)}] (3.28)

where ca and φ are, respectively, the adhesion and the angle of friction between a tire anda terrain. These values comprise another set of terrain-tire system constants which can bedetermined from the tire segment plate traction test.


Figure 3.19. The various forces acting on a tire climbing up sloped terrain during driving action.

As shown in Figure 3.19, the axle load W , the driving torque Qd and the apparent effectivedriving force Td0 of a rolling tire which is climbing up a sloping terrain of angle β duringdriving action, can be expressed by the following equation:

W cos β = BR

[∫ θf

θC

{σ(θ) cos θ + τ(θ) sin θ} dθ +∫ θC

−θC

pg cos{δ(θ)} cos θ dθ

+∫ −θC

−θr

{σ(θ) cos θ + τ(θ) sin θ} dθ

](3.29)

Qd = BR2

{∫ θf

θC

τ(θ) dθ +∫ θC

−θC

τ(θ) cos θ dθ +∫ −θC

−θr

τ(θ) dθ

}(3.30)

Td0 = BR

[∫ θf

θC

{τ(θ) cos θ − σ(θ) sin θ} dθ +∫ θC

−θC

τ(θ) cos θ dθ

+∫ −θC

−θr

{τ(θ) cos θ − σ(θ) sin θ} dθ

](3.31)

Additional to the above equation, the apparent effective driving force Td0 can be furtherexpressed as the difference between the thrust Thd , the compaction resistance Rcd and theslope resistance W sin β as follows:

Td0 = Thd − Rcd − W sin β (3.32)

where

Thd = BR∫ θr

−θr


Rcd = BR

{∫ θf

θC

σ(θ) sin θ dθ +∫ −θC

−θr

σ(θ) sin θ dθ

}(3.34)

102 Terramechanics

In this case, the amount of slip sinkage is not considered. The thrust Thd is set equal to theintegral of τ(θ) cos θ acting in the moving direction of the tire. The compaction resistanceRcd is the land locomotion resistance that occurs as a consequence of the formation of arut of static depth s0. In magnitude it is equal to the integral of σ(θ) sin θ and it acts in theopposite direction to the moving direction of the tire.

Further, the driving force can be set equal to Qd /R. For flat terrains, the horizontalcomponent (Qd /R)h can be expressed as the sum of a number of components. These arethe total compaction resistance Lcd , e.g. the land locomotion resistance calculated takinginto consideration the amount of slip sinkage, and the actual effective driving force Td , asshown in the previous Eq. (2.31).

The amount of slip sinkage ss at an arbitrary point X on the terrain can be calculatedfrom the amount of slippage of soil js given in the previous Eq. (2.53) as follows:

ss = c0

Nf∑n=1


{( n

Njs)c2 −

(n − 1

Njs

)c2}

+c0

Nr∑n=Nf

{p(θn) cos δn}c1

{( n

Njs)c2 −

(n − 1

Njs

)c2}

+c0

N∑n=Nr

{p(θn) cos (θn − δn)}c1

{( n

Njs)c2 −

(n − 1

Njs

)c2}

(3.35)

where

θn = n

N(θf + θr) δn = tan−1

{τ(θn)

σ(θn)

}

Nf = Nθf − θC

θf + θCNr = N

θf + θC

θf + θr

In the above equation, the coefficient c0, and the indices c1, c2 are the terrain-tire systemconstants obtained from the tire segment plate slip sinkage test.

The rut depth s associated with the passage of a tire can be calculated by use of theprevious Eq. (2.55) taken with the values of the static amount of sinkage s0 and u0 givenin the previous Eqs. (3.20), (3.21) and the amount of slip sinkage ss given in the aboveequation. Thence, in a manner similar to that employed to generate the previous Eq. (2.56),the total amount of compaction Lcd can be calculated using the following equation:

Lcd = k1ξB∫ s

0zn1 dz (3.36)

Next, the actual effective driving force Td can be expressed as the difference betweenthe tangential component of the driving force (Qd /R)s in the direction of the surface ofterrain, and the total compaction resistance Lcd (given in the above equation) and the sloperesistance W sin β as follows:

Td =(

Qd

R

)s

− Lcd − W sin β (3.37)


In Figure 3.19, the product of the normal ground reaction N to the sloped terrain surfaceand the apparent amount of eccentricity ed0 can be seen to be equal to the integral of theelemental ground pressure distribution force couples taken over the range θ = −θr to θ = θf .More particularly, this integral is comprised of the sum of:(i) the product of the normal component of the resultant pressure to the sloped terrain

surface p(θ) RB cos θ cos(θ − δ) dθ acting on an elemental area RB cos θ dθ at an arbi-trary point on the contact part BC and AD to the terrain and with moment arm lengthR sin θ and

(ii) the product of the normal component of the resultant pressure to the sloped terrainsurface p(θ)RB cos θC cos δ dθ acting on an elemental area RB cos θC dθ at an arbitrarypoint on the contact part AB and with a moment arm length R cos θC .

Thence, the apparent amount of eccentricity ed0, without considering any slip sinkage, canbe calculated as:

N = W cos β (3.38)

ed0 = BR2

W cos β

{∫ θf

θC

p(θ) cos θ cos (θ − δ) sin θ dθ

+ cos2 θc

∫ θC

−θC

p(θ) cos δ dθ +∫ −θC

−θr

p(θ) cos θ cos(θ − δ) sin θ dθ

}(3.39)

However, to calculate the actual amount of eccentricity ed , it is necessary to modify theabove equation to allow for the effects of slip sinkage.

The effective input energy E1 supplied by the driving torque Qd acting on a tire may beequated to the sum of a number of output energies: namely a compaction energy componentE2 – associated with the development of a rut under the tire, a slippage energy componentE3 – associated with the development of shear deformation at the interface between tireand terrain, an effective driving force energy component E4 e.g. the traction work requiredto draw another vehicle and a potential energy component E5 that comes into play whenthe vehicle or wheel is on a slopped terrain. Since the system must be in energy balancethe following equation applies:

E1 = E2 + E3 + E4 + E5 (3.40)

When the radius vector of the tire rotates from an entry angle θf at the beginning of thecontact part of the tire to the terrain to the central angle θf + θr , the moving distance of thetire is equal to R(θf + θr)(1 − id) during the rotation of an arbitrary point on the peripheralcontact part of R(θf + θr). The individual energy components that arise during the rotationof the tire can be computed as follows:

E1 = Qd(θf + θr) (3.41)

E2 = LcdR(θf + θr)(1 − id) (3.42)

E3 = Qd(θf + θr)id (3.43)

E4 = TdR(θf + θr)(1 − id) (3.44)

104 Terramechanics

E5 = WR(θf + θr)(1 − id) sin β (3.45)

Further, the amount of energy produced or consumed per unit of time can be calculatedfrom knowledge of the peripheral speed Rω of the wheel and the moving speed V of thesystem in the direction of an arbitrarily sloped terrain. For these calculations, the followingequations can be used.

E1 = Qdω =(

Qd

R

)s

V

1 − id(3.46)

E2 = LcdRω(1 − id) = LcdV (3.47)

E3 = Qdωid =(

Qd

R

)s

idV

1 − id(3.48)

E4 = TdRω(1 − id) = TdV (3.49)

E5 = WRω(1 − id) sin β = WV sin β (3.50)

Substituting the previous Eq. (3.37) into the above equations, the requisite energyequilibrium balance of Eq. (3.40) can be confirmed.

The optimum effective driving force Tdopt during driving action can be defined as theeffective driving force Td at the optimum slip ratio idopt when the effective driving forceenergy E4 takes a maximum value for a constant circumferential speed Rω. Additionally,the tractive efficiency Ed can be calculated from the previous Eq. (2.68).

(2) During braking action

Similar to the driving action case, a simplified shape of deformation of a tire can be adoptedas shown in Figure 3.20.

Here the central angle of the straight line section of the flat part AB is 2θC , and the centralangles of the arc portions BC and AD are θf − θC and θr − θC respectively. The angle θ

is the central angle of the radius vector and is defined to be measured counter clockwisefrom the vertical line of the tire. The parameter R is the original radius of the tire and B isthe width of the tire.

Let us now consider the force balance that exists between the axle load W , the brakingtorque Qb and the effective braking force Tb during braking action and the distributions ofthe contact pressures σ(θ) and τ(θ) that must act on the contact parts of the tire and theterrain.

The amount of slippage j = jb(θ) on the respective contact parts of the arc BC, the straightline segment AB and the arc AD can be determined by use of the previous Eq. (2.71), i.e.

For θC � θ � θf

jb(θ) = R

{(θf − θ) − 1

1 + ib(sin θf − sin θ)

}(3.51)

For −θC � θ � θC

jb(θ) = jb(θC) + R(sin θC − sin θ)ib

1 + ib(3.52)


Figure 3.20. Distributions of amount of slippage jb(θ) and ground reaction a(θ), τ(θ) of contact partof tire (during braking action).

For −θr � θ � −θC

jb(θ) = jb(−θc) + R∫ −θC

θ

(1 − 1

1 + ibcos θ

)dθ

= jb(−θc) − R

{(θC + θ) − 1

1 + ib(sin θC sin θ)

}(3.53)

For the central angle θ = α at the maximum value jb(θ) = jp, the shear resistance τ(θ) takesa positive value for 0 � jb(θ) � jp in the range of θf � θ �α and takes a negative value forjb(θ)< jq in the range of α � θ �−θr as mentioned previously in Eq. (2.89).

On the other hand, the normal stress σ(θ) takes a positive value for the whole range of thecontact area. The angle δ between the applied resultant stress p(θ) and the radial directionof the contact part of tire is determined as δ = tan−1{τ(θ)/σ(θ)}.

The magnitude of the sinkage s0 just under the central axis of the tire and the amountof rebound u0 of the terrain just after the passage of the tire, respectively, are given in theprevious Eqs. (3.20) and (3.21).

106 Terramechanics

The applied resultant stress p(θ) can be determined given an elemental length of thecomponent of the rolling locus d(θ) – as obtained from previous Eq. (2.85) – in the direc-tion of the applied stress q(θ), the coefficient of modification ξ considering the dynamicmechanics of the tire, and the angle of the effective braking force ζ = tan−1(Tb/W cosβ) asin the following equations:

For θmax � θ � θf

p(θ) = k1ξ{d(θ)}n1

cos (ς − θ + δ)(3.54)

For θC � θ < θmax


cos(ς − θ + δ)(3.55)


p(θ) = k1ξ{d(θmax)}n1 − k2{d(θmax) − d(θC)}n2

cos(ζ − θ + δ)(3.56)


p(θ) = ||k1ξ{d(θmax)}n1 − k2{d(θmax) − d(θC)}n2 − k2[d(θmax) − {d(θC) + d ′(θ)}]n2 ||cos(ζ − θ + δ)

(3.57)

where θmax is the central angle when p(θ) takes a maximum value, and d ′(θ) is the length ofthe component of the rolling locus integrating the previous Eq. (2.85) from θ = θ to θ = θC .

Thence, the distribution of normal stress σ(θ) can be expressed as:

σ(θ) = p(θ) cos{δ(θ)} (3.58)

Assuming that the applied resultant stress is constant, e.g. p(θ) = pg , then σ(θ) is given as:

σ(θ) = pg cos{δ(θ)} (3.59)

Similarly, the distribution of shear resistance τ(θ) can be calculated as discussed in relationto the previous Eq. (2.89). By substituting the amount of slippage jb(θ) given in Eqs. (3.51),(3.52) and (3.53) into the previous Eqs. (2.8) and (2.9) the following results are obtained.

For the traction state: 0 � jb(θ) � jp

τ(θ) = {ca + σ(θ) tan φ}[1 − exp{−ajb(θ)}]For the untraction state: jq < jb(θ) < jp

τ(θ) = −{ca + σ(θ) tan φ}[1 − exp{−a[ jq − jb(θ)]}] (3.60)

For the reciprocal traction state: jb(θ) � jq

τ(θ) = −{ca + σ(θ) tan φ}[[1 − exp [ − a{ jq − jb(θ)}]]]Thence, the axle load W , the braking torque Qb and the apparent braking force Tb0 acting onthe tire which is descending a slope of angle β during braking action as shown in Figure 3.21


Figure 3.21. The various forces acting on a tire descending down sloped terrain during braking action.

can be computed by use of the following equations.

W cos β = BR

[∫ θf

θC

{σ(θ) cos θ + τ(θ) sin θ} dθ +∫ θC

−θC

pg cos{δ(θ)} cos θ dθ

+∫ −θC

−θr

{σ(θ) cos θ + τ(θ) sin θ} dθ

](3.61)

Qb = BR2

[∫ θf

θC

τ(θ) dθ +∫ θC

−θC

τ(θ) cos θ dθ +∫ −θC

−θr

τ(θ) dθ

](3.62)

Tb0 = BR

[∫ θf

θC

{τ(θ) cos θ − σ(θ) sin θ}dθ +∫ θC

−θC


+∫ −θC

−θr

{τ(θ) cos θ − σ(θ) sin θ} dθ

]

In the above equation, the apparent effective braking force Tb0 can be expressed as thedifference between the drag Thb, and the sum of the compaction resistance Rcb and theslope resistance W sin β as follows:

Tb0 = Thb − Rcb − W sin β (3.64)

where

Thb = BR∫ θf

−θr


Rcb = BR

[∫ θf

θC

σ(θ) sin θ dθ +∫ −θC

−θr

σ(θ) sin θ dθ

](3.66)

In this case, the amount of sinkage ss is not considered. The drag Thb is given as theintegral of τ(θ)cos θ acting in the opposite direction to the moving direction of the tire and

108 Terramechanics

the compaction resistance Rcb e.g. the land locomotion resistance for the static amount ofsinkage s0 is given as the integral of σ(θ)sin θ acting in the same direction.

The braking force is given as Qb/R. As shown in the previous Eq. (2.83), the horizon-tal component of the braking force (Qb/R)h on a flat terrain equals the sum of the totalcompaction resistance Lb {e.g. the land locomotion resistance calculated considering theamount of slip sinkage} and the actual effective braking force Tb.

The amount of slip sinkage ss at an arbitrary point X on the terrain can be calculated usingEq. (3.35) by substituting the amount of slippage js of the soil into Eq. (2.100). The depthof rut s that develops after the passage of the tire can be calculated by use of the previousEq. (2.55) by substituting in the static amount of sinkage s0, the amount of rebound u0

given in the previous Eqs. (3.20) and (3.21) and the amount of slip sinkage ss. Thence,the total compaction resistance Lcb can be determined, in the manner already mentioned inconnection with the previous Eq. (2.103), as follows:

Lcb = k1ξB∫ s

0zn1 dz (3.67)

The actual effective braking force Tb can be calculated as the difference between the tan-gential component of the braking force in the direction of the sloped terrain surface (Qb/R)s,and the sum of above mentioned total compaction resistance Lcb and the slope resistanceW sin β. This relation can be represented as:

Tb =(

Qb

R

)s

− Lcb − W sin β (3.68)

When the amount of slip sinkage is not considered, the normal ground reaction N to thesloping terrain and the apparent amount of eccentricity eb0 may be given by:

N = W cos β (3.69)

eb0 = BR2

W cos β

{∫ θf

θC


+ cos θc

∫ θC

−θC

p(θ) cos δ dθ +∫ −θC

−θr


}(3.70)

For full accuracy however, the actual amount of eccentricity eb should be modified to allowfor the effects of slip sinkage.

The effective input energy E1 supplied from the braking torque Qb acting on a tire can beequated to the sum of a number of output energy components: namely, a compaction energycomponent E2 – associated with the development of a rut under the tire, a slippage energycomponent E3 – associated with the development of shear deformation at the interfacebetween tire and terrain, an effective braking force energy component E4 e.g. the brakingwork and a potential energy component E5 that comes into play on sloping terrains.

When the radius vector of the tire rotates from an entry angle θf at the beginning of thecontact part of the tire-terrain interface through to a central angle θf + θr , the moving dis-tance of the tire is equal to R(θf + θr)/(1 + ib) during a rotation of R(θf + θr). The individualenergy components, either generated or absorbed during the rotation of the tire, can thenbe calculated as follows:

E1 = Qb(θf + θr) (3.71)


E2 = LcbR(θf + θr)

1 + ib(3.72)

E3 = Qb(θf + θr)ib1 + ib

(3.73)

E4 = TbR(θf + θr)

1 + ib(3.74)

E5 = WR(θf + θr) sin β

1 + ib(3.75)

Following this, the production or expenditure or energy per unit time for each componentcan be calculated given knowledge of the peripheral speed for the wheel Rω and the movingspeed V in the direction of the sloping terrain through use of the following equations:

E1 = Qbω =(

Qb

R

)s

V (1 + ib) (3.76)

E2 = LcbRω

1 + ib= LcbV (3.77)

E3 = Qbωib1 + ib

=(

Qb

R

)s

ibV (3.78)

E4 = TbRω

1 + ib= TbV (3.79)

E5 = WRω sin β

1 + ib= WV sin β (3.80)

Substituting the previous Eq. (3.68) into the above Eq. (3.79), the requisite energyequilibrium balance can be confirmed.

The ‘optimum effective braking force’ Tbopt during braking action can be defined as theeffective braking force Tb at the ‘optimum skid’ ibopt when the effective input energy |E1|takes a maximum value for a constant tire moving speed V . Thence, the braking efficiencyE can be calculated using the previous Eq. (2.114).

3.4 KINEMATIC EQUATIONS OF A WHEEL

So far, the traffic performances of a rigid wheel and a flexible tired wheel during drivingand braking action have been analysed assuming a constant peripheral speed Rω and aconstant moving speed of tire V . These analyses were developed on the assumption thatthese wheels roll at a uniform speed and at a constant slip ratio or skid. In the more common,non-steady state condition however, the driving and braking torques Qd(t), Qb(t), and theeffective driving and braking forces Td(t), Tb(t) vary with time t. Similarly, the slip ratioid and the skid ib will vary with time in correspondence with variations in the peripheralspeed Rω and the moving speed V of the wheel.

Since the coefficients of driving and braking resistance µd , µb vary with time, thesewheels will roll generally with either an accelerated speed or with a decelerated speed.

110 Terramechanics

In this section, the fundamental kinematic equations of the straight forward motion andthe rotational motion of the wheel will be analysed and developed for the pure rolling modeand for the driving and braking modes of operation of a wheel.

To begin these analyses, it is noted that the kinematic equations for the pure rollingstate of the wheel can be developed in terms of the moment of inertia I and gravitationalacceleration g as follows:

Idω

dt= −(µrlr + er)W (3.81)

W

g· dV

dt= µrW − Tr (3.82)

where µr(< 0) is the coefficient of rolling friction, lr is the normal distance between theapplication point of the ground reaction Br acting in the direction of terrain surface and thecentral longitudinal axis as shown in the previous Figure 2.15. The factor er is the amount ofeccentricity of the normal ground reaction N whilst Tr is the rolling resistance of the wheel.

When a wheel is drawn under conditions such that the rolling resistance Tr is alwayscontrolled to be equal to µrW , the rate of change of velocity dV /dt becomes zero in theabove equation and as a consequence the wheel rolls with a uniform speed under a constantskid ir(< 0). Then, the angular velocity ω of the wheel becomes constant so that dω/dt = 0and the condition er = − µrlr(µr < 0) is satisfied {as shown in the previous Eq. (2.80)}.

When a wheel is drawn by a Tr that is greater than µrW , dV /dt becomes less thanzero and consequently the wheel rolls with a decelerated speed. Since Br = µrW increaseswith increments of skid, dω/dt becomes less than zero in the above equation. Under thesecircumstances, both the moving speed V and the angular velocity ω decrease and the wheeldecelerates at the same time.

On the other hand, when the wheel is drawn by a Tr that is smaller than µrW , the factordV /dt becomes larger than zero and as a consequence the wheel rolls with an acceleratedspeed. Since Br = µrW decreases with decrements of skid, the parameter dω/dt becomeslarger than zero in the above equation so that both the moving speed V and the angularvelocity ω increase and the wheel accelerates at the same time.

As a next step, the kinematic equation of the tire during driving action can be establishedfor an applied driving torque as follows:

Idω

dt= Qd(t) − (µd ld + ed)W (3.83)

W

g· dV

dt= µdW − Td(t) (3.84)

where µd is the coefficient of driving resistance, ld is the normal distance between theapplication point of the ground reaction Bd acting in the direction of terrain surface andthe central longitudinal axis as shown in the previous Figure 2.8(c). The factor ed is theamount of eccentricity of the normal ground reaction N and Td(t) is the actual effectivedriving force. Additionally, µd is a factor that can be generally calculated from the previousEqs. (2.25) and (3.37) and can be predicted for sloping terrains by the following expression:

µd = Bd

W=(

Qd

RW

)s

− Lcd

W− sin β (3.85)


When a wheel is drawn so that the effective driving force Td(t) is always controlled such thatBd = µdW , the parameter dV /dt becomes zero in the above Eq. (3.84) and as a result thewheel rolls with a uniform speed at a constant slip ratio id . Thence, the driving torque Qd(t)becomes equal to (µd ld + ed)W so that dω/dt = 0, e.g. the angular velocity ω becomes aconstant value. This results in the following equation;

ω = V

R(1 − id)(3.86)

where µd , ld and ed are given as the function of slip ratio id as mentioned previously.When Qd(t) increases or decreases {rather than having a value of (µd ld + ed)W for aconstant moving speed V}, dω/dt takes on a positive or negative value with the result thatthe circumferential speed of the wheel Rω either increases or decreases. Similarly, sincethe values of µd , ld , ed and Td(t) vary with increment or decrement of slip ratio id the forceequilibrium of the system can be maintained. On the other hand, when Td(t) increases ordecreases, rather than having a value of µdW , for a constant peripheral rotational speedRω, dV /dt takes on negative or positive values so that the moving speed of the wheeldecreases or increases. Under these circumstances, moment equilibrium is maintainedthrough corresponding variation of µd , ld , ed and Qd(t). Likewise, when Qd(t) and Td(t)increase or decrease simultaneously, µd , ld , ed vary with V and Rω and force and momentbalances are continuously maintained.

Supplementally, the kinematic equations of a tire during braking action can be set-up fora braking torque Qb as follows:

Idω

dt= Qb(t) − (µblb + eb)W (3.87)

W

g· dV

dt= µbW − Tb(t) (3.88)

where µb(<0) is the coefficient of braking resistance, lb is the normal distance betweenthe application point of the ground reaction Bb acting in the direction of the terrain surfaceand the central longitudinal axis as shown in the previous diagram, Figure 2.15(b). Theparameter eb is the amount of eccentricity of the normal ground reaction N and Tb(t) isthe actual effective braking force. Typically, the factor, µb can be calculated from theprevious Eqs. (2.81) and (3.68). Predicted values can be obtained for sloping terrains usingthe following equation

µb = Bb

W=(

Qb

RW

)s

− Lcb

W− sin β (3.89)

When a tire is braked under conditions such that the effective braking force Tb(t) is alwayscontrolled to have a magnitude Bb = µbW , then dV /dt becomes zero in the above Eq.(3.88) and as a consequence one can see that the wheel rolls with an uniform speed with aconstant amount of skid ib. Under these circumstances, the braking torque Qb(t) becomes(µblb + eb)W , so that dω/dt = 0 e.g. the angular velocity ω takes on a constant value asdescribed by the following equation:

Rω = (1 + ib)V (3.90)

112 Terramechanics

Figure 3.22. Friction force F acting on tire and slip velocity at cornering state.

where µb, lb and eb are functions of the amount of skid as mentioned previously. WhenQb(t) increases or decreases rather than having a value equal to (µblb + eb)W for a constantmoving speed V , dω/dt takes on a positive or negative value and as a consequence thecircumferential speed of the wheel Rω increases or decreases. Because the values of µb,lb, eb and Tb(t) vary with increment or decrement in the amount of skid ib the system forceequilibrium can be maintained. Alternatively, when Tb(t) increases or decreases ratherthan having a value µbW for a constant peripheral rotational speed Rω, dV /dt takes anegative or positive value such that the moving speed of the wheel decreases or increases.Accordingly, moment equilibrium can be maintained by variations in µb, lb, eb and Qb(t).Under conditions where Qb(t) and Tb(t) increases or decreases simultaneously, µb, lb, eb

vary with V and Rω such that the force and moment balances are always maintained.The curves ofµd−id and µb−ib of a tire running on a soft terrain depend principally on the

soil properties, the materials that comprise the tread rubber of the tire and the roughness ofthe tire surface. Curves of these parameters can be determined either from a simulation anal-ysis that uses experimentally determined terrain-tire system constants or else can be directlydetermined from actual land locomotion tests on a tire during driving and braking action.

3.5 CORNERING CHARACTERISTICS

When a tire turns in some direction to the straight-line direction, the tire rolls but has associ-ated with this process a lateral slippage at some angle β to the moving direction. This angleβ is called the ‘angle of lateral slippage β of the tire’. It is defined as the angle between thelongitudinal direction and the moving direction of the tire as illustrated in Figure 3.22.

In the left-hand and right-hand diagrams (a) and (b), the directions of the friction forceF acting on a tire during cornering action in driving and braking state are shown relative tothe corresponding velocity vectors. For a velocity vector u in the moving direction of thetire and a velocity vector Rω in the rotational direction, a slippage velocity vector vs canbe expressed as Rω + u. The angles between F , u, vs and the longitudinal direction of thetire can be designated by the variables δ, β, γ respectively. The sense of these variables isdefined such that clockwise angles represent positive values.

The longitudinal slip ratio or skid ilon of a tire can be defined:For the driving state where 0 � u cos β < Rω as:

ilon = 1 − u cos β

Rω(3.91)


Figure 3.23. Relationship between longitudinal and lateral coefficients of friction µlon and µlat forvarious longitudinal slip ratio ilon and angle of lateral slippage β [24].

For the braking state where Rω � u cos β as:

ilon = Rω

u cos β− 1 (3.92)

In the above equations, ilon takes on a positive value for the driving state and negative onefor the braking state.

In a similar vein, a lateral slip ratio ilat for a tire can be defined as:

ilat = −u sin β

u= − sin β (3.93)

for positive values of the lateral slippage angle β. In this case, the lateral slip ratio ilat

takes a negative value as a consequence of the counterclockwise left turning of the tire.If the coefficient of friction µ is divided into a longitudinal component µlon and a lateralcomponent µlat , the relationship between these components can be expressed as a groupof elliptical curves. Figure 3.23 [23, 24] shows a group of such curves developed aroundthe two parameters of longitudinal slip ratio ilon and angle of lateral slippage β. It can beseen here that the lateral coefficient of friction µlat takes a maximum value of (µlat)max atµlon = 0 for each value of β. Further, this value can be expressed as an exponential functionof β as in the following equation:

(µlat)max = MAX(µ1at){1 − exp(−k1β)} (3.94)

where MAX(µlat) is the maximum value of µlat for the whole range of the values of β

and ilon.Similarly, the longitudinal coefficient of friction µlon takes a maximum value of (µlon)max

at µlat = 0 for each value of ilon and can be expressed as an exponential function of ilon asfollows:

(µlon)max = MAX(µlon){1 − exp(−k2ilon)} (3.95)

114 Terramechanics

where MAX (µlon) is the maximum value of µlon for the whole range of the values of β

and ilon.In the above two equations, the coefficients k1 and k2 vary with the soil properties and

the tread pattern of the tire.Further, the group of elliptical curves that are shown by the solid lines in the figure, can

be expressed as follows:

{µlat

(µlat)max

}2

+{

µlon

MAX (µlon)

}2

= 1 (3.96)

and another group of elliptical curves, shown by the dotted lines, can be expressed asfollows{

µlat

MAX (µlat)

}2

+{

µlon

(µlon)max

}2

= 1 (3.97)

An approximately linear relationship can be established between the coefficient of frictionµ and the slip ratio i of a tire running on a hard terrain.

Namely:

µlon = clonilon (3.98)

µlat = clat ilat (3.99)

Likewise, the resultant force F applied to a tire can be divided into a longitudinal componentFlon and a lateral component Flat as follows:

Flon = µlonW (3.100)

Flat = µlatW (3.101)

Further, the resultant force F can be sub-divided into a drag force FB acting in the movingdirection of tire and a cornering force FC acting in the centrifugal direction as shown inFigure 3.24:

FB = Flon cos β − Flat sin β = (clonilon cos β − clat ilat sin β)W (3.102)

FC = Flon sin β + Flat cos β = (cilon sin β + clat ilat cos β)W (3.103)

In this diagram, the symbol M is the self-aligning torque that occurs as a result of the pointof application of the cornering force FC moving from the center of the tire towards the rearof the tire.

Figure 3.25 shows the shape of the contact part of the tire during cornering action in thedriving state. Initially, the tread of the tire touches the terrain at the beginning contact pointl and moves along the line l–m during the rolling of the tire.

In this situation, the cornering force FC increases with the lateral deformation of thetire until it reaches a maximum frictional resistance when slip occurs reversely due to anelastic deformation of the rubber tread. Then, the tire tread kicks away from the point m andrecovers suddenly to the end contact point n. The interval l–m is referred to as the cohesive


Figure 3.24. Forces acting on tire.

Figure 3.25. Shape of contact area cornering state (during driving action).

zone and that of m–n as the slippage zone. In this case, the drag force FB can be calculatedfrom the driving or braking torque and the land locomotion resistance, and the corneringforce FC can be computed as the sum of the static frictional force and the dynamic frictionalforce acting on the contact part of the tire. Sakai [25] has developed a detailed mechanicalmodel of the process of transitioning from the static frictional zone to the dynamic one.

As mentioned previously, the application point of FC deviates to the rear part of the tireand away from the central line of the tire which is normal to the terrain surface. From thisa self-aligning torque Mdevelops. Gough [26] measured the relationship between FC andM for a 165-SR-13 tire with air pressure of 167 kPa for various values of axle load W . Theresults he obtained are shown in Figure 3.26.

The value of M takes a maximum value for the range of β = π/45 ∼ π/30 rad and afterthat it decreases gradually. The value of M increases linearly with increments in FC fora range of small values of β but tends to decrease for a range of large values of β. TheNewmatic trail NT is the amount of eccentricity that exists between the application point

116 Terramechanics

Figure 3.26. Relationship between cornering force Fc and self aligning torque M [26].

of FC and the central line of the tire normal to the terrain surface. This parameter can becalculated as follows:

NT = M

FC(3.104)

The ratio of the cornering force FC to the angle of slippage β for the range of small valuesof β, i.e. FC /β is designated as the ‘Cornering power’. The ratio of the cornering power tothe axle load of the tire, i.e. FC /Wβ is designated the ‘Coefficient of cornering’. Both theseterms may be used to describe the cornering characteristics of a tire.

3.6 DISTRIBUTION OF CONTACT PRESSURE

In this section, some illustrative results relative to measured contact pressure distributions atthe tire-terrain interface are presented. Krick [13] measured the distribution of the contactpressure that existed on the surface of a pneumatic tire that was rolling on a loose accu-mulated sandy loam terrain, of water content 19%. He obtained his measurements by useof a three dimensional transducer. Figure 3.27 shows the distributions of contact pressuree.g. the normal stress σ and the shear resistance τ measured for a tire during driving stateoperations. The axle load was 5.34 kN and the slip ratio 10%.

In this case, the shear resistance distribution shows a triangular shape which increaseslinearly with distance from the beginning point of the contact of the tire with the terrain.On the other hand, the normal stress follows a parabolic shape. In the diagram, the curvedenoted by the symbol 1 refers to the results measured at the center of the tire width whilstcurve 4 refers to the results measured at the edge part of the tire. It is noted that the normalstress shows some stress concentration at the edge part of the tire.

Figure 3.28 shows the distributions of contact pressure of a tire of axle load 5.34 kNoperating in a driving state with a slip ratio of 40%. Under such conditions of large slipratio values, it is observed that the distributions of normal stress and shear resistance show


Figure 3.27. Contact pressure distributions under tire (slip ratio 10% during driving action) [13].

Figure 3.28. Contact pressure distributions acting on tire (slip ratio 40% during driving action) [13].

118 Terramechanics

Figure 3.29. Relationship between ratio of maximum normal stress σmax, maximum and minimumshear resistance τmax, τmin to axle load W and slip ratio s.

a somewhat flattened shape at the center but they also show a triangular shape in the entryand exit contact regions of the tire.

In somewhat similar studies, Oida et al. [27] measured values for the distributions ofnormal stress σ, longitudinal and lateral shear resistance τ acting on the contact part ofa tire running on a standard sand. They did their work systematically for various combi-nations of longitudinal slip ratio s and angle of lateral slippage β. The sand had an angle


of internal friction of 38 degrees. The tire was of type 4.50-5-4 PR and the air pressurewas 49 kPa.

Figure 3.29 presents the results of this work in terms of three normalised parameters i.e.the maximum normal stress divided by the axle load σmax/W, the maximum longitudinalshear resistance divided by the axle load τmax/W, the minimum longitudinal shear resistancedivided by the axle load τmin/W.These normalised values are plotted against the longitudinalslip ratio s for various values of angle of lateral slippage β.

Some experimental relations that best-fit this data are as follows.σmax

W= −4.9 × 10−4s + 0.1653 (3.105)

τmax

W= 8.174 × 10−5s + 0.0308 (3.106)

τmin

W= 4.284 × 10−4s + 8.356 × 10−3 (3.107)

From these results, it is observed that the angle of lateral slippage β does not significantlyaffect the distributions of normal stress σ and shear resistance τ. The data also shows thatthe ratio of the angle that develops a maximum normal stress σmax to the angle developing amaximum shear resistance τmax for a specific entry angle can be expressed as a polynomialfunction of the longitudinal slip ratio s.

3.7 SUMMARY

In this chapter, the work of Chapter 2 on rigid drums has been extended to cover elastic anddeformable drums and short cylinders operating over hard and deformable terrains. Becausethese systems are hard to handle mathematically and because three dimensional effectscannot reasonably be ignored for most rubber tired vehicles, this chapter has put togethera mix of empirical and analytical functions that can be used in tired-vehicle behaviourprediction situations.

In this chapter, basically the same set of drum/terrain modelling situations that werecovered in Chapter 2 with rigid wheels have been revisited with flexible wheels.

REFERENCES

1. Kommandi, G. (1975). Determination of the Peripheral force for Pneumatic Tyres Rolling onConcrete Surfaces. Proc. 5th Int. Conf. ISTVS, Michigan, U.S.A., Vol. 2. pp. 567–588.

2. Muro, T. & Enoki, M. (1983). Characteristics of Friction of OR Tyre at Cornering Site. Memoirsof the Faculty of Engineering, Ehime University, Vol. X, No.2, pp. 285–297, (In Japanese).

3. Yong, R.N., Boonsinsuk, P. & Fattah, E.A. (1980). Tyre Flexibility and Mobility on Soft Soils.J. of Terramechanics, 17, 1, 43–58.

4. Fujimoto, Y. (1977). Performance of Elastic wheels on Yielding Cohesive Soils. J. ofTerramechanics, 14, 4, 191–210.

5. Forde, M.C. (1978). An Investigation into Rubber Wheel Mobility on London Clay and CheshireClay. Proc. 6th Int. Conf. ISTVS, Vienna, Austria, Vol. 2. pp. 587–642.

6. Yong, R.N., Boonsinsuk, P. & Fattah, E.A. (1980). Tyre Load Capacity and Energy Loss withRespect to Varying Soil Support Stiffness. J. of Terramechanics, 17, 3, 131–147.

7. Yong, R.N., Fattah, E.A. & Boonsinsuk, P. (1978). Analysis and Prediction of Tyre–SoilInteraction and Performance using Finite Elements. J. of Terramechanics, 15, 1, 43–63.

120 Terramechanics

8. Abeels, P. (1981). Studies of Agricultural and Forestry Tyres on Testing Stand, Basis for DataStandardizations. Proc. 7th Int. Conf. ISTVS Calgary, Canada, Vol. 2, pp. 439–453.

9. Fujimoto, Y. (1977). Performance of Elastic Wheels on Yielding Cohesive Soil. J. ofTerramechanics, 14, 4, 191–210.

10. Harlow, S., Krutz, G., Liljedahl, J.B. & Parsons, S. (1981). Relation between Tractor Tyre Wearand Lug Forces. Proc. 7th Int. Conf. ISTVS, Calgary, Canada. Vol. 2. pp. 645–662.

11. Muro, T. (1983). Characteristics of Wear life of Heavy Dump Truck Tyre. Proc. of JSCE,No. 336, pp. 149–157, (In Japanese).

12. Turnage, G.W. (1978). A Synopsis of Tire Design and Operational Considerations Aimed atIncreasing In-Soil tire Drawbar Performance. Proc. 6th Int. Conf. ISTVS Vienna, Austria, Vol. 2,pp. 759–810.

13. Krick, G. (1969). Radial and Shear Stress Distribution under Rigid Wheels and Pneumatic TiresOperating on Yielding Soils with Consideration of tire Deformation. J. of Terramechanics, 6,3, 73–98.

14. Oida, A. & Triratanasirichai, K. (1988). Measurement and Analysis of Normal, Longitudinaland Lateral Stresses in Wheel-Soil Contact Area. Proc. 2nd Asia-Pacific Conf. ISTVS, Bangkok,Thailand. pp. 233–243.

15. Söhne, W. (1953). Druckverteilung im Boden und Bodenverformung unter Schleppereifen.Glundlagen der Landtechnik, Heft 5, pp. 49–63.

16. Muro, T. (1985). Compaction Phenomena due to Vehicle Transportation. Tsuchi-to-Kiso, Vol. 33,No. 9, pp. 33–38, (In Japanese).

17. Bolling, I. (1981). Verticalspannungsverteilungen im Boden unter Luftreifen, Proc. 7th Int. Conf.ISTVS Calgary, Canada. Vol. 2, pp. 497–529.

18. Bekker, M.G. Prediction of Design and Performance Parameters in Agro-Forestry Vehicles.National Research Council of Canada, Report No. 22880.

19. Schwanghart, H. (1991). Measurement of Contact Area, Contact Pressure and Compaction underTires in Soft Soil. J. of Terramechanics, 28, 4, 309–318.

20. Karafiath, L.L. & Nowartzki, E.A. (1978). Soil Mechanics for Off-Road Vehicle Engineering.pp. 355–427, Transtech Publications.

21. Blaszkiewicz, Z. (1990). A Method for the Determination of the Contact area between a Tyreand the Ground. J. of Terramechanics, 27, 4, 263–282.

22. Wong, J.Y. (1989). Terramechanics and Off-Road Vehicles. pp. 214–241. Elsevier.23. Grecenko, A. (1975). Some Applications of the Slip and Drift Theory of the Wheel. Proc. 5th

Int. Conf. ISTVS, Michigan, U.S.A. Vol. 2, pp. 449–472.24. Crolla, D.A. & El-Razaz, A.S.A. (1987). A Review of the Combined Lateral and Longitudinal

Force Generation of Tyres on Deformable Surfaces. J. of Terramechanics, 24, 3, 199–225.25. Sakai, H. (1969). Theoretical Considerations of the Effect of Braking and Driving Force on

Cornering Force. Automotive Engineering, 23, 10, 982–988, (In Japanese).26. Gough, V.E. (1954). Cornering Characteristics of Tyres. Aut. Engr., Vol. 44, No. 4.27. Oida, A., Satoh, A., Itoh H. & Triratanasirichai, K. (1991). Three-Dimensional Stress Distribu-

tions on a Tire-Sand Contact Surface. J. of Terramechanics, 28, 4, 319–330.

EXERCISES

(1) Suppose that an off-road-tire of radius of R = 1.2 m is standing on a hard terrain andthat it is sustaining an axle load of W = 100 kN. Under these conditions the amount ofdeformation of the tire f is observed to be 3 cm. The radius of the tread crown r of thetire is 30 cm. Assuming that the tire deforms only at the contact tire/terrain interface andthat the contact pressure equals the tire air pressure, calculate the contact area A and thelengths of the major axis 2a and the minor axis 2b of the ellipse.

(2) Suppose that an off-the-road tire of radius R = 1.20 m is running on a hard terrain, in apure rolling state, and that it sustains an axle load of W = 100 kN. If the effective radius


of rotation Re0 is 1.16 m and the coefficient of rolling friction µr is 0.20, calculate theeffective braking force Tb and the amount of eccentricity er of the ground reaction.

(3) Imagine that an off-the-road tire of radius of R = 1.30 m is running during driving actionon a hard terrain whilst it is sustaining an axle load of W = 100 kN. If the applied drivingtorque Qd is 100 kNm, the effective radius of rotation Re0 is 1.25 m, and the coefficient ofrolling friction µd is 0.20, calculate the coefficient of driving resistance µd , the effectivedriving force Td , and the amount of eccentricity ed of the ground reaction.

(4) Imagine that an off-the-road tire of radius of R = 1.20 m, width of B = 0.5 m is runningduring driving action on a weak sandy terrain and whilst it is doing this it is sustainingan axle load of W = 20 kN. If the height of the tire H is 0.6 m, the gradient of coneindex G of the terrain is 12 N/cm3, and the amount of deformation δ of the tire is 3.0 cm,calculate the effective tractive effort T20 and the tractive efficiency η20 at a slip ratio ofid = 20% from the wheel mobility number Ns.

(5) Consider an off-the-road tire of radius of R = 1.0 m which is running during brakingaction on a hard terrain while it is sustaining an axle load of W = 100 kN. Given anapplied braking torque Qb of 80 kNm, an effective radius of rotation Reb is 0.96 m, anda coefficient of rolling friction µr = 0.15, calculate the coefficient of braking resistanceµb, the effective braking force Tb and amount of eccentricity eb of the ground reaction.

(6) An off-the-road tire of radius of R = 1.2 m and width of B = 0.8 m is running during driv-ing action on a soft clayey terrain while sustaining an axle load of W = 10 kN. Supposethat the height of the tire H is 0.6 m, the cone index C of the terrain is 15.6 N/cm2, and theamount of deformation δ of the tire is 5 cm. Calculate the effective tractive effort T20 andthe tractive efficiency η20 at a slip ratio of id = 20% from the wheel mobility number Nc.

(7) A 13.6/12-28 tire of diameter D = 1310 mm and width B = 345 mm is standing on a looseaccumulated sandy loam terrain of water content w = 15%. It is sustaining an axle loadof W = 17.9 kN. The tire air pressure p is 150 kPa. Does the tire behave as a rigid wheelor as a flexibly tired wheel?

(8) A tire of radius of R = 30 cm, width of B = 10 cm sustaining the axle load W = 3000 N isdescending a slope of β = π/36 rad in a pure rolling mode. The depth of rut produced is5 cm and values of terrain-tire system constants equal to k1 = 4.5 N/cmn1+1, n1 = 0.809and ξ = 1 have been obtained from a plate loading test. Calculate the effective brakingforce Tb in this case.

(9) An off-the-road tire is climbing up a slope of angle β during driving action. Demonstratethat the effective input energy per second E1 is equal to the sum of the output energies persecond, i.e. equal to the summation of the sinkage deformation energy E2, the slippageenergy E3, the effective drawbar pull energy E4, and the potential energy E5.

(10) Suppose that a tire, which is sustaining an axle load of W = 2.5 kN, is cornering duringdriving action on a sandy terrain at a slip angle of β = π/18 rad and that a longitudinalslip ratio of ilon = 15% prevails. Calculate the cornering force FC and the drag force FB

in this case.

Chapter 4

Terrain-Track System Constants

Track-laying vehicles, as they manifest in modern machines such as bulldozers, army tanks,tracked mobile-cranes or snowmobiles comprise a major class of land-locomotion device.These vehicles interact with the ground though a track belt structure that can be thoughtof as existing in two quite distinctive forms – a rigid track form and a flexible track form.Because the interaction between a track belt and a terrain is fundamentally different in therigid and the flexible belt types, the overall structure of a machine’s track belt system willdictate the traffic performance of the machine on construction sites or on other types ofon-road or off-road application.

A multiroller-type track belt [1] is an example of the essentially fully-rigid track belttype. In this arrangement, the movement of the track bush that mutually connects severaltrack links is fully constrained by a structural girder. As a result of this constraint, noupper or lower movement of the track plates that are connected to the track links can occur.The distribution of contact pressure under a rigid track belt system is uniform and as aconsequence rigid tracked vehicles can typically develop large drawbar pulls – especiallyon soft terrains.

On the other hand, most common bulldozers and tractors are examples of an essentiallyflexible track belt undercarriage system. In this arrangement, upper and lower movementsof the track plates between several road rollers running on the track link can occur. As well,right and left hand lateral movements of the road rollers can take place. The distributionof contact pressure under such flexible track belt systems follows a wavy distribution withstress concentrations existing just under the road rollers. As a consequence of this, flexibletrack vehicles can be used for operations on rough terrain.

In the study of tracked machinery, a variety of parameters that collectively will be referredto as terrain-track system constants can be used to describe and predict the interactions thatoccur between a track belt structure and a terrain. The terrain-track constants have beendeveloped through considerations of basic soil mechanics.

The terrain-track system constants are constants that can model the relations betweencontact pressure and amount of sinkage through use of a model-track-plate loading andunloading test. As well, the relations that exist between shear resistance, contact pressureand amount of slippage as well as the relations that exist between the amount of slip sinkage,contact pressure and the amount of slippage of the soil can be derived from a model-track-plate traction test. These constants are dependent on the structure of the track belt, the ratioof the grouser pitch to height and the properties of the terrain materials.

In what follows, the terrain-track system constants for silty loam, decomposed weatheredgranite soil and snow covered terrains will be investigated. As well, the size effect of themodel-track on the constants will be studied.

124 Terramechanics

4.1 TRACK PLATE LOADING TEST

The relations between the contact pressure p acting on a track belt and the static amountof sinkage s0 vary with the terrain properties, the structure of the track belt and the size ofthe model-track. Because of these multiple factors it is necessary to carry-out loading testson model-tracks equipped with variously sized track plates of given grouser height H andgrouser pitch Gp for particular terrains.

The experimental measurements that cover this type of test generally take the follow-ing form.

For p ≤ p0, s0 ≤ H

p = k1sn10 (4.1)

For p > p0, s0 > H

p = p0 + k2(s0 − H )n2 (4.2)

where p0 = k1H n1 .

The two equations cover the situations where (a) the grouser penetrates the surface but thetrack-plane has not landed and (b) the grouser has penetrated and contact with the bottomplane of the model-track has occurred. As well, it is necessary to execute an unloading teston the model-track element from a start-point of an arbitrary amount of sinkage sp. Thistest is required so as to determine the relations between the contact pressure p and the staticamount of sinkage s0 in the unloading phase since a process of unloading takes place afterthe pass of a road roller on a flexible track belt. For the unloading case, the form of theexperimental equations is as follows:

For p ≤ p0, s0 ≤ H

p = k1sn1p − k3(sp − s0)n3 (4.3)

For p > p0, s0 > H

p = p0 + k2(sp − H )n2 − k4(sp − s0)n4 (4.4)

where the coefficients of sinkage k1, k2, k3 and k4, and the indices of sinkage n1, n2, n3 and n4

are the set of terrain-track system constants that are determined from the model-track-plateloading test.

4.2 TRACK PLATE TRACTION TEST

From a traction test on a model-track-plate, the relationship that exists between the shearresistance acting at a track-terrain interface, the contact pressure p and the amount ofslippage j can be experimentally determined. Also, the relationship between the amountof slip sinkage ss, the contact pressure p and the amount of slippage js of a soil can bedetermined.

The shear resistance τ of a soil that develops under a track belt can not be expressed simplyby a cohesion and angle of internal friction expression that satisfies the Mohr-Coulomb’s

Terrain-Track System Constants 125

Figure 4.1. (a) Type of exponential function (A Type). (b) Hump type relationship (B Type) betweenshear resistance τ and amount of slippage j.

failure criterion [2]. The reason for this is that also involved is the failure pattern of the soilthat is induced by the structure of track belt and the degree of mobilization of the shearstrength of the soil that occurs due to the operation of various slippage processes.

In general, the shear resistance τ under a track belt can be expressed as a function ofthe apparent cohesion mc, the apparent angle of shear resistance mf , the contact pressurep and the amount of slippage j. As shown in Figure 4.1, the function can be classed intotwo representative types that reflect particular terrain properties. Thus, one can have anexponential type function (Type A) that passes through the origin and is asymptotic to amaximum value τmax. The other type of expression is a Hump type function (Type B) whichexhibits a distinctive peak value.

For loose accumulated sandy soils, remolded soft clayey soils and for normal consolidatedclays, Janosi-Hanamoto [3] and others have proposed the following Type A function:

For 0 ≤ j ≤ jp (traction state):

τ = (mc + mf p){1 − exp(−aj)}

126 Terramechanics

For jq < j < jp (untraction state):

τ = τp − k0( jp − j)n0 (4.5)

For j ≤ jq (reciprocal traction state):

τ = −(mc + mf p)[1 − exp{−a(jq − j)}]where jp is an arbitrary value for the amount of slippage j at the beginning of the untractionstate, τp is the corresponding shear resistance at j = jp, and jq is the value of j when theshear resistance τ becomes zero in the untraction process. Further, the constant a is thecoefficient of deformation divided by τmax.

For a firmly compacted sandy soil and a hard terrain, the following Type B function hasbeen proposed by Oida [4]:

τ = fm · p

∥∥∥∥∥∥1 −√

1 − fm/fn exp[(j/jm) log

{1 + fs/fm ·

(√1 − fs/fm − 1

)}]√

1 − fm/fs (1 − 2fs/fm) + 2fs/fm − 2

∥∥∥∥∥∥×∥∥∥1 − exp

[( j/jm) log

{1 + fs/fm ·

(√1 − fm/fs − 1

)}]∥∥∥ (4.6)

where fm is the ratio of the residual shear resistance and the contact pressure τres/p, fs is theratio of the maximum shear resistance and the contact pressure τmax/p, and jm is the amountof slippage corresponding to the maximum shear resistance τmax. The above equation canalso be applied for non-adhesive asphalt and concrete pavement.

For an overconsolidated clay, the following B type function was proposed by Bekker [5]:

τ = (mc + mf p)

×exp

{(−K2 +

√K2

2 − 1)

K1 j

}− exp

{(−K2 −

√K2

2 − 1)

K1 j

}

exp{(

−K2 +√

K22 − 1

)K1 jm

}− exp

{(−K2 −

√K2

2 − 1)

K1 jm

} (4.7)

K1 = 1

jm·

log(

−K2 −√

K22 − 1

)√

K22 − 1

where K1, K2 are the coefficients of deformation of the soil and jm is the amount of slippagecorresponding to the maximum shear resistance.

The above mentioned coefficient e.g. mc, mf and a in Eq. (4.5), fm, fs and jm in Eq. (4.6),and K1, K2 and jm in Eq. (4.7) are the terrain-track system constants determined from thetrack-plate traction test.

Some further relations between the amount of slip sinkage ss and the amount of slippageof soil js may also be deduced from the track plate traction test. The slip sinkage of thetrack belt occurs due to a dilatation phenomenon [6] that appears in concert with theshear deformation of the soil under a track belt. It is also due to the scratching effects ofthe grousers. In general, the amount of slip sinkage ss measured at the rear end of the


model-track-plate is given as a function of contact pressure p and amount of slippage js asfollows:

ss = c0pc1 jc2s (4.8)

where the constant c0 as well as the indices c1, c2 vary with the terrain properties and thestructure of the track belt. These terrain-track system constants need to be determined froma series of track-plate traction tests.

4.3 SOME EXPERIMENTAL RESULTS

4.3.1 Effects of variation in grouser pitch–height ratio

The terrain-track system constants vary very considerably with the physical form of aparticular track belt. Specifically, the ratio of the grouser pitch Gp to the grouser height Hseems to be a very important parameter in determining the magnitude of the overall tractiveaction that develops under a track belt.

From traction tests [7] carried out on a model-track-plate which was equipped withstandard T shaped rubber grousers and which was operating on a sandy terrain, it wasobserved that the traction force took on a maximum value when the Gp/H ratio was in therange 3 ∼ 4.

Figure 4.2 shows the tractive effort development mechanism for a model-track-platewhich is equipped with standard T shaped grousers. With increasing amounts in slippage jof the model-track-plate, slip lines grow in the soil between the side planes of the grousers.As a consequence, tractive effort develops due to progressive failure along the slip lineswhich in turn develops shear resistance.

When the ratio of Gp/H takes on values less than around 2, the profile of the slip linein the soil between the grousers approaches the form of a straight line connecting each ofthe tips of the grousers. Under these conditions, the maximum tractive effort T of a model-track-plate having n individual grouser elements can be determined for a track width B,a contact pressure p, a coefficient of earth pressure at rest K0, and an angle of soil shearresistance tan φ and represented by the following equation:

T = nGpBp tan φ

[1 + 2HK0

B

](4.9)

Figure 4.2. Mechanism of occurrence of slip lines and tractive effort.

128 Terramechanics

Figure 4.3. Relation between coefficient of traction T /W and grouser pitch-height Gp/H .

In actuality, the value of T can be presumed to decrease gradually with decrements in thegrouser pitch Gp, because the base area of the grouser itself becomes relatively large withdecreasing Gp and also because the frictional resistance between the grouser tip and thesoil becomes less than the internal shear resistance of the soil.

When the ratio of Gp/H takes on values larger than around 4, the slip line in the soilbetween the grousers becomes a passive failure line which consists of a logarithmic spiralslip line and a straight slip line. This failure line develops in front of each grouser. Underthese conditions, the maximum tractive effort T of a model-track-plate having n individualgrouser elements can be approximated (for a coefficient of passive earth pressure Kp) bythe following expression:

T = nHBKpp + nGpHpK0 tan φ (4.10)

In this case, the vertical load W acting on the model-track-plate is equal to the productnGpBp. Also, by dividing the above equation by W, the following expression is obtained:

T

W= HKk

Gp+ HK0 tan φ

B(4.11)

where T /W decreases with increases in Gp.Figure 4.3 shows the relations between the ratios T /W and Gp/H . In the Figure the dotted

line A represents a modified Eq. (4.9). The dotted line B represents Eq. (4.11).Under real world conditions, however, a somewhat lesser value of T /W than that sug-

gested by the A or B lines occurs. The ratio T /W takes a maximum value at some value ofGp/H as shown by the solid line. It is noted however that this optimum grouser pitch–heightratio (Gp/H )opt varies with the particular properties of the terrain.

Likewise to the above, from traction tests [8] for a model-track-plate equipped with aset of standard T shaped steel grousers and operating on a remolded silty loam terrain, itwas observed that the shear resistance of the soil τ70 at an amount of slippage j = 70 cmtook on a maximum value at Gp/H ∼= 3.2. Figure 4.4 shows the relationships between τ70

and Gp/H for three magnitudes of contact pressure p = 7.1, 10.1, and 12.3 kPa. The model-track-plate of width B = 20 cm was equipped with 7 individual T shaped steel grousers ofheight H = 3.2 cm. The water content and the cone index of the silty loam terrain weremeasured to be approximately 30% and 31 kPa respectively.


Figure 4.4. Relationship between shear resistance τ70 at amount of slippage 70 cm and grouserpitch–height ratio Gp/H .

Figure 4.5. Relationship between shear resistance τ, amount of sinkage s and amount of slippagej(Gp = 10.2 cm).

Table 4.1. Terrain-track system constants (silty loam terrain).

Gp (cm) 5.1 10.2 20.4k2 (N/cmn2+2) 10.50 10.17 9.85n2 0.395 0.395 0.395mc (kPa) 10.00 9.020 8.330mf 0.471 0.620 0.401a (1/cm) 0.140 0.150 0.130c0 (cm2c1−c2+1/Nc1 ) 0.139 0.100 0.105c1 0.860 0.855 0.754c2 0.425 0.466 0.403

Figure 4.5 plots the relations between the shear resistance τ, the amount of sinkage s = s0+ss

and the amount of slippage j for a grouser pitch of Gp = 10.2 cm. Table 4.1 gives a summaryof all the terrain-track system constants, including the model-track-plate loading test results,for three values of grouser pitch Gp = 5.1, 10.2 and 20.4 cm.

130 Terramechanics

Width B = 25 cm, grouser pitch Gp = 14.6 cm,grouser height H = 6.5 cm, base length L = 2,3,4.5 cm,trim angle α = π/6 rad

Figure 4.6. Shape and dimension of track model plate.

Figure 4.7. Relationship between contact pressure p and static amount of sinkage s0.

4.3.2 Results for a decomposed granite sandy terrain

In this section some results for both a track plate loading and a traction test are presentedfor a model-track-plate equipped with equilateral trapezoidal rubber grousers that is actingon a loose accumulated decomposed weathered granite sandy terrain.

The test decomposed granite sandy soil has a specific gravity of 2.66, an average grainsize of 0.78 mm, a coefficient of uniformity of 2.0, a maximum density of 1.68 g/cm3,and a minimum density of 1.31 g/cm3. The air dried decomposed granite sandy soil wasdeposited by free-falling the material from a height of 1.0 m and filling a bin uniformlyuntil a depth of 30 cm was attained. The test was undertaken in a large soil bin of length270 cm, width 90 cm and height 60 cm.

The relative density of the test terrain prepared in this way was 44.0%, the water contentwas 2.38%, the dry density was 1.44 g/cm3 and the cone index was 95.1 kPa. The terrainresembled a loose accumulated sandy soil of the kind that would have been generated bythe loose spreading action of the blade of a bulldozer or motor grader. The rubber grousersin this case were made of a rubber of spring hardness of Hs = 62. As depicted in Figure 4.6,the model-track-plate was equipped with a set of equilateral trapezoidal grousers of trimangle α = π/6 rad, base length L = 2, 3, 4 and 5 cm, and of height H = 6.5 cm arranged toa pitch Gp = 14.6 cm.

In one series of studies, the effects of the base length of rubber grouser on the terrain-track system constants were investigated. Figure 4.7 presents the results of such a series ofmodel-track-plate loading tests. From the diagram, it can be seen that the magnitude of thestatic amount of sinkage s0 for a given contact pressure p increases with a decrease in thebase length L.


Figure 4.8. Relationship between shear resistance τ and amount of slippage j (L = 4 cm).

Figure 4.9. Relationship between amount of slip sinkage Ss and amount of slippage js (L = 4 cm).

For a corresponding model-track-plate traction test, the measured relationship between theshear resistance τ and the amount of slippage j for a parameter value of L = 4 cm is givenin Figure 4.8. Similarly, the relationship between the amount of slip sinkage ss and amountof slippage js for L = 4 cm is shown in Figure 4.9. The τ–j relations shown belong to a typeof exponential function as discussed in relation to the previous Eq. (4.5) where mc equalszero, and the mf and a values vary with L.

Figure 4.10 shows a plot of the relationships between mf , a and base length L. From this,it is noted that mf takes a maximum value at L = 3 ∼ 4 cm and a takes a minimum value atL = 3 cm. On the other hand, Figure 4.11 shows a plot of the relationships between amountof slip sinkage ss and base length L. Under these conditions it can be seen that ss takes amaximum value at L = 4 cm.

Table 4.2 presents a summary of all the various terrain-track system constants obtainedfrom the above experiments.

4.3.3 Studies on pavement road surfaces

In this section, some experimental results for a track-plate traction test are presented.In this case results were obtained for a model-track-plate equipped with rectangular and

132 Terramechanics

Figure 4.10. Relationship between constants mf , a and base length L.

Figure 4.11. Relationship between amount of slip sinkage Ss and base length L ( js = 73 cm,p = 17.76 kPa).

Table 4.2. Terrain-track system constants (decomposed granite soil).

L (cm) 2 3 4 5

k1 (N/cmn1+2) 8.526 11.76 17.05 21.56n1 0.866 1.188 1.082 1.202mc (kPa) 0 0 0 0mf 0.769 0.824 0.822 0.788a 0.244 0.233 0.260 0.272c0 (cm2c1−c2+1/Nc1 ) 1.588 0.996 0.566 0.448c1 0.075 0.264 0.453 0.436c2 0.240 0.274 0.295 0.330

equilateral trapezoidal rubber grousers acting on concrete and asphalt pavement roadsurfaces. The roughness of the asphalt pavement road surface was greater than that ofthe concrete pavement, and the traction tests were executed under air-dried conditions.

In both the cases, the pavement road surface did not deform during the track plateloading test and hence the terrain-track system constants k1, n1 and c0, c1, c2 must be


Figure 4.12. (a) Relationship between shear resistance τ and amount of slippage j on concrete pave-ment road (equivalent trapezoidal grouser, L = 3 cm). (b) Relationship between shear resistance τand amount of slippage j on asphalt pavement road (equivalent trapezoidal grouser, L = 3 cm).

zero. The rubber grousers used in the trials were made of rubber of a spring hardness ofHs = 62. Two kinds of rectangular grouser of height H = 6.5 cm and base length of L = 3 and5 cm respectively, and four kinds of equilateral trapezoidal grousers of trim angle π/6 rad,height H = 6.5 cm and base length L = 2, 3, 4 and 5 cm respectively were employed. Themodel-track-plate of width of 25 cm was fitted with a set of five rectangular or equilateraltrapezoidal grousers arranged to a pitch Gp = 14.6 cm and Gp/H = 2.25.

Under these conditions, several aspects of the influence of the type of grousers andthe base length of the grousers on the terrain-track system constants have been studied.Figure 4.12(a) graphs the discovered relationship that exists between the shear resistanceτ and the amount of slippage j of the equilateral trapezoidal grousers.

The grousers had a base length of L = 3 cm and were tested on a concrete pavement roadunder various contact pressures p. Figure 4.12(b) shows the same data obtained for tests onan asphalt pavement.

From these tests, it can be seen that both of the τ–j relations belong to the Hump typecategory as discussed previously in relation to Eq. (4.6). Given this hump-type relation, theterrain-track system constants fs, fm and jm can thence be determined. Figure 4.13 showsa plot of the relationship between fs and base length L for several combinations of types ofgrousers operating on both concrete and asphalt pavement road surfaces.

Figure 4.14 gives the relationships between fm and L for the same combinations of grouserand road pavement type. Since equilateral trapezoidal grousers are structurally stronger thanthe rectangular ones, the constants fs and fm for the equilateral trapezoidal grousers yieldhigher values than those for the rectangular grousers.

134 Terramechanics

Figure 4.13. Relationship between constant fs and base length L.

Figure 4.14. Relationship between constant fm and base length L.

While both the values of fs and fm show some characteristics changes with base length L,the values for asphalt pavement are generally larger than those for concrete pavements.

Table 4.3 summarises all the terrain-track system constants obtained from this particularseries of trials.

4.3.4 Scale effects and the model-track-plate test

In discussions to date, a variety of model tests have been introduced which have sought toclarify the mechanics of the real-world interaction between a soil and a piece of constructionmachinery through the use of models. However it is well recognised in engineering circlesthat in many situations it is difficult to apply the results of a model test directly to a fullscale problem because of size-effect problems. To investigate the nature of scale effects,a number of geometrically similar model tests were tested to estimate, for example, the


Table 4.3. Terrain-track system constants (road pavement).

Road pavement Type of grouser L (cm) fs fm jm (mm)

Concrete Rectangular 3 0.918 0.856 6.175 0.944 0.823 5.76

Equilateral trapezoid 2 1.028 0.887 8.873 1.193 0.972 4.844 1.113 0.992 6.255 1.138 0.959 3.75

Asphalt Rectangular 3 0.920 0.916 10.875 0.980 0.949 4.89

Equilateral trapezoid 2 1.134 1.010 6.623 1.217 1.026 5.614 1.200 1.081 6.865 1.259 1.037 3.93

Table 4.4. Modified list of variables.

Variable Symbol Basic dimensions

Lengths λ, λa LForces R, R1 FSoil properties α FaLb

αi Fai Lbi

bulldozing resistance of the blade of a bulldozer. From such tests, the actual behaviourof a blade can be estimated from a model by application of the principles of dimensionalanalysis [9,10].

Muro and Kawahara [11] confirmed the existence of a size effect in the estimationof actual tractive resistances and amounts of sinkage through use of the model-track-platetraction test. The study was done to investigate the trafficability of a tracked vehicle runningon a soft terrain. In the study, the researchers considered a distorted model condition (seelater explanation) in which the same soil was used for the model test as was the actualterrain upon which the full scale construction machinery was operating. The researchersdeveloped a quantitative approach to estimate the actual tractive resistance and amount ofsinkage as a function of a ratio of geometrical similarity by use of dimensional analysisconsideration and through use of the well known Buckingham-Pi principles of analysiswhich uses a number of non-dimensional parameters or groups called � terms.

Schafer et al. [12] took the general variables as shown in Table 4.4 to cover the similarityproblem of soil-machine system constants. They assumed that the variables of accelerationand so on that included a dimension of time could be neglected in the field of terramechanicssince agricultural and construction machinery move comparatively slowly. From the list ofselected variables, they developed the following non-dimensional � terms;

∏1

= R

α(1/a)λ(b/a)(4.12)

136 Terramechanics

∏l= R1

R(4.13)

∏q

= λq

λ(4.14)

∏αi

= αi

α(ai/a)λbi − (bai/a)(4.15)

where �l is a term which includes the factor R which should be estimated, �l is a termcomprised of a ratio between several factors, �q is likewise a non-dimensional term com-prised of the ratio between several dimensions, �αi is a term that includes the soil constants.The factor n is a ratio of similarity e.g. nR = R/Rm, nR1 = R1/R1m, nλ = λ/λm, nα = α/αm,nλq = λq/λqm. The subscript m expresses the model relationship. Under these conditions,the design relationships which need to be satisfied for similitude are as follows:

➀ nR = n(1/a)a n(b/a)

λ (4.16)

➁ nRl = nR (4.17)

➂ nλq = nλ (4.18)

➃ nai = n(ai/a)α nbi − (bai/a)

λ (4.19)

For the above l + q + I + 1 conditional equations, the number of unknown factors arel + q + I + 3 in which the two values of nλ and nα can be given arbitrarily. Here, thecondition which is usually considered is nα = nαi = 1, if α is the same as αi. It is noted,however, that it is additionally necessary to satisfy either of the next conditions to establishcondition ➃.

➄ αi is a non-dimensional quantity e.g. ai = bi = 0.➅ α has the same dimension as that of αi.➆ The dimensions of α and αi satisfy the equation bi = bai/a.

As the condition ➃ is distorted when these conditions are not satisfied, a distorted modelsystem should be considered. A coefficient of distortion βi for the soil constants can bedefined as follows:

βi =∏

αim∏αi

= nbi − (bai/a)λ = nsi

λ (4.20)

si = bi − bai

a(4.21)

That is, βi can be expressed as a function of the ratio of similarity nλ.Moreover, for the distorted model system, it is necessary to define the following coeffi-

cient of estimation δ of � terms. This collection should include the force which should beestimated:

δ =∏

1∏1m

= f(∏

2,∏

3, . . . ,∏

c

)f(∏

2m,∏

3m, . . . ,∏

cm

) (4.22)


When �1 and the distorted �αi are expressed by a power function, the factor δ can then bepresented as follows:

δ = nλs

s = −∑

eisi (4.23)

where ei is the index of �αi in the relation between �1 and �αi .These results show that, when a model test is carried out by use of the same soil as that

of the actual terrain, an estimation of the actual behaviour of a prototype from a model testresult is possible even if the soil constants belong to the distorted system. δ is a function ofsize ratio nλ only, and the index s is constant for a given soil-machine system.

As an example of the use of a traction test employing a model-track-plate to investigatethe size effect, several dimensional analytical results are presented here.

The data given are for the maximum tractive resistance Fmax and the amount of sinkagezm of a standard T shaped model-track-plate running on a weak silty loam terrain. Thesilty loam terrain was prepared by remolding the loam uniformly and then adjusting it towater contents of 30, 35 and 40%. The standard T shaped model-track-plate employed isshown schematically in Figure 4.15. In total, twelve different kinds of model-track-platewith grouser pitch–height ratio of Gp/H = 1.5, 3.0 and 4.5 and with size ratios of nλ = 1,2, 4 and 8, as shown in Table 4.5, were employed. In these tests, the contact pressure p wasalternately selected as 0.98, 2.94, 4.90, 6.86 and 9.80 kPa.

Figure 4.15. Standard T shaped track model plate.

Table 4.5. Shape and dimension of track model plate.

Track Track length Track width Grouser height Grouser pitch Gp Gp/Hmodel D (cm) B (cm) H (cm) (cm)

I 29.0 5.0 1.0 4.5 4.529.0 5.0 1.5 4.5 3.029.0 5.0 3.0 4.5 1.5

II 58.0 10.0 2.0 9.0 4.558.0 10.0 3.0 9.0 3.058.0 10.0 6.0 9.0 1.5

III 116.0 20.0 4.0 18.0 4.5116.0 20.0 6.0 18.0 3.0116.0 20.0 12.0 18.0 1.5

IV 232.0 40.0 8.0 36.0 4.5232.0 40.0 12.0 36.0 3.0232.0 40.0 24.0 36.0 1.5

138 Terramechanics

In these tests, the width B (cm), the length D (cm), the grouser pitch Gp (cm) and the heightH (cm) of the model-track-plate were selected as the variables in the size dimension whilstthe maximum tractive resistance Fmax(N) and the load W (N) were selected as the variablesin the force dimension. The undrained shear resistance cu (kPa) (FL−1) and the weight perunit volume γ (kN/m3) (FL−3) were selected as the variables in the soil constants.

For a value of Fmax which needs to be estimated, the experimental results can be arrangedin the following � terms:∏

1= Fmax

cuB2

∏2= W

cuBD= p

cu

∏3= Gp

H

∏4= γB

cu

(4.24)

For the independent variable �1 and the dependent variables �2, �3 and �4, all theexperimental data may be analysed through use of multi-regression analysis. In relation tothe results, �1 can be expressed as the product of �2, �3 and �4 as shown in the nextequation:∏

1= 13.07

∏0.496

2

∏−0.586

3

∏−0.333

4(4.25)

Substituting the previous Eq. (4.24) and the value γ = 18.2 kN/m3 into above equation,Fmax can be approximated as follows:

Fmax = 4.97c0.837u p0.496B1.667

(Gp

H

)−0.586

(4.26)

When the soil constants of model test are the same as those of prototype, ncu = nγ = 1, andsi = −1, s = −0.333 for the indices a = 1, b = −2, ai = 1, bi = −3. Then the coefficient ofdistortion β and the coefficient of estimation δ can be formulated as:

β =∏

4m∏4

= n−1B (4.27)

δ =∏

1∏1m

= n−0.333B (4.28)

Thence, substituting the value of �1 given in Eq. (4.24) into Eq. (4.28), the ratio of (Fmax)p

of the prototype to (Fmax)m of the model can be expressed as:

(Fmax)p

(Fmax)m= n1.667

B (4.29)

Where the value of the index is greater than, say, 2.0 then there is no size effect. In thiscase, the reason for the occurrence of the size effect is considered to be the fact that thelength of the slip line in front of the top grouser and the side area between grousers doesnot increase linearly with increments in grouser height H because the amount of sinkageof the model-track-plate is not in proportion to the grouser height H .

Next, we can define the amount of sinkage zm as the amount of sinkage measured at thecenter of the model-track-plate when the tractive resistance takes a maximum value Fmax.


For a value of zm, which needs to be estimated, the experimental results can be arranged inthe following � terms:∏′

1= zm

H

∏2= p

cu∏3= Gp

H

∏′4= γH

cu

(4.30)

Additionally, to satisfy the design condition ➃ in this experiment, it is necessary to controlthe soil constants so that they are uniform in the direction of depth. Taking the results of amulti-regression analysis for all the experimental data,

∏′1 can be expressed as the product

of∏

2,∏

3 and∏′

4 as shown in the next equation:

∏′1

= 0.705∏0.559

2

∏0.312

3

∏′−0.046

4(4.31)

Substituting the previous Eq. (4.30) into above equation, zm can be approximated as follows:

zm = 0.617c−0.513u p0.559H 0.954

(Gp

H

)0.312

(4.32)

As the term∏′

4 is the distorted � term, the coefficient of distortion β and the coefficientof estimation δ can be expressed, in a similar way to the case of Fmax, as follows:

β =∏′

4m∏′4

= n−1H (4.33)

δ =∏′

1∏′1m

= n−0.046H (4.34)

Thence, substituting the value of �1 given in the previous Eq. (4.30) into the aboveEq. (4.34), the amount of sinkage of a prototype (zm)p can be expressed by use of thatof the model (zm)m as follows:

(zm)p

(zm)m= n0.954

H (4.35)

Since the index nH is less than 1.0, it is evident that the relative amount of sinkage �′1 of

the model is larger than that of the prototype. This occurs because, for the small size ofthe model-track-plate, the relative amount of sinkage �′

1 becomes large due to the shortaverage moving distance of the soil when the sandwiched soil material between the grousersis displaced sideways, while, for the large size of model-track-plate, the amount of sinkageis restrained due to difficulty in the lateral or sideways movement of the sandwiched soilelements between the grousers.

In summary of all the above, it is clearly evident that there are some size effects on boththe values of Fmax and zm in the traction test of the model-track-plate.

Next, let us look at some test results that study several of the size effects that operate inthe tractive performance of a model-track-plate equipped with eight equilateral trapezoidal

140 Terramechanics

Figure 4.16. Equivalent trapezoidal track model plate.

Figure 4.17. Relationship between contact pressure p and static amount of sinkage s0 for various sizeratios N .

rubber grousers. The cross section of the test plate is as depicted in Figure 4.16. The testswere carried out on an unsaturated weak terrain. Four kinds of geometrically similar model-track-plates having a size ratio of N = 1, 2, 3 and 4 were utilised in the plate loading andtraction test.

The dimensions of the model-track-plate for N = 1 are as follows: contact length D =41 cm, width B = 12.5 cm, grouser height H = 1.5 cm, grouser pitch Gp = 4.5 cm, trimangle α = π/6 rad and base length of grouser L = 0.5 cm. The grouser pitch–height ratiofor each size ratio was set equal to 3.0 to maximize tractive resistance. The unsaturated weakterrain was prepared by remolding a silty loam to a water content of 30%. This material wasthen filled into a large soil bin (540 × 150 × 60 cm) until a depth of 40 cm was achieved.The measured cone index of the terrain was 31 kPa and the degree of saturation 96.9%.

Figure 4.17 shows the results of carrying out a plate loading test on a model-track-plate. The values of s0 (cm) for a constant value of contact pressure p (kPa) increasedwith increment of N because the range of influence of the pressure bulb extends due tothe increasing total load applied to the model-track-plate. Figure 4.18 shows the relationsbetween the terrain-track system constants k1(N/cmn1+1), n1, k2(N/cmn2+1), n2 given inprevious Eqs. (4.1), (4.2) and the size ratio N . The factors k1, k2 decrease with increasingN , but n1, n2 increase with N .


Figure 4.18. Relationship between constants k1, n1, n2, and size ratio N .

Figure 4.19. Relationship between shear resistance of soil τ and amount of slippage j for various sizeratios N .

The resulting, experimentally derived, equations are as follows:

k1 = 1.50 × 10N −1.192 (4.36)

k2 = 5.69N −0.015 (4.37)

n1 = 7.30 × 10−1N 0.183 (4.38)

n2 = 7.30 × 10−1N 0.098 (4.39)

To illustrate these concepts, Figure 4.19 shows the results of a model-track-plate tractiontest i.e. the relationships between shear resistance τ (kPa) and amount of slippage j (cm),for various values of N . The value of τ decreases with increasing values of N , because the

142 Terramechanics

Figure 4.20. Relationship between constants mc, mf , a and size ratio N .

bulldozing resistance that acts in front of the model-track-plate tends to decrease due to adecreasing amount of sinkage of the front part of the model-track-plate accompanied by anincreasing angle of inclination of the plate.

Figure 4.20 shows the relationships between the terrain-track system constants mc (kPa),mf , and a (1/cm) given in the previous Eq. (4.5) and the size ratio N . Each value decreaseswith increasing N . The experimental equations that best fit this data are:

mc = 6.04N −0.282 (4.40)

mf = 4.13 × 10−1N −0.137 (4.41)

a = 4.35 × 10−1N −0.824 (4.42)

Figure 4.21 shows, as an illustrative study, the relationship between the amount of slipsinkage ss (cm) of a model-track-plate and the amount of slippage js (cm) for various valuesof N . The factor ss increases with increasing values of contact pressure p and amounts ofslippage js because of an increasing bulldozing action and volume change in the soil dueto the dilatancy phenomenon that occurs during shear action. Additionally it is noted thatss increases with increasing values of N because the influence zone of the pressure bulbpenetrates more deeply.

Figure 4.22 shows the relations between the terrain-track system constants c0, c1, and c2

given in the previous Eq. (4.8) and the size ratio N .

c0 = 4.19 × 10−2N 1.349 (4.43)


Figure 4.21. Relationship between amount of slip sinkage s0 and amount of slippage js.

Figure 4.22. Relationship between constants c0, c1, c2 and size ratio N .

c1 = 1.15N −0.324 (4.44)

c2 = 2.40 × 10−1N 0.331 (4.45)

As to these results, both the values of c0, c2 increase with increases in N . In contrast,though, the value of c1 decreases with N .

144 Terramechanics

Figure 4.23. Traction test of standard T shaped track model plate.

Figure 4.24. Relationship between contact pressure p and static amount of sinkage s0 for circularplate and track model plate.

4.3.5 Snow covered terrain

In this section we present some experimental test results relative to the undertaking of aloading test and a traction test on snow. In this case, the apparatus consisted of a model-track-plate equipped with standard T shaped grousers. The terrain was a snow covered onecomposed of newly fallen winter snow at an outside temperature of 0 ◦C.

The depth of the snow was 20 cm. The average density of the newly fallen snow ρ0

was 0.228 g/cm3 and the water content of the wet snow was 1.02%. Both a circular plateof diameter of 7.0 cm and a model-track-plate of contact pressure of D = 29 cm, widthB = 5 cm, grouser height H = 1.5 cm, and grouser pitch of Gp = 4.5 cm were used in theloading trials. A cross section of the apparatus employed is shown in Figure 4.23.

Figure 4.24 gives the results of the loading tests for the circular plate and the model-track-plate respectively. It can be seen in this graph that, for a constant contact pressure,


Figure 4.25. Relationship between shear resistance of snow τ and amount of slippage j for variouscontact pressures.

the amount of sinkage of the model-track-plate becomes larger than that of the circularplate. This is presumed to be due to the penetration action of the grouser itself.

Following the straight loading test, a series of traction tests on the model-track-plate werecarried out at various levels of constant pressure p = 0.42, 0.84, 1.27 and 2.55 kPa. Toassure a traction-speed of 3.7 mm/s, the model-track-plate was pulled by a wire and winch.

Figure 4.25 shows the results for the shear resistance τ (kPa) of the snowy terrain andthe amount of slippage j (cm) plotted for various levels of contact pressure p (kPa). As isvery evident from the graphs, each curve in this series follows a Hump type relationship inwhich the shear resistance τ decreases rapidly after a peak. This is due to a brittle failurewithin the snow material after it passes a maximum value. For each dynamic slip sinkagerelation in the traction test, it is observed that the total amount of sinkage s (i.e. the sum ofthe static amount of sinkage s0 and the dynamic amount of slip sinkage ss) increases rapidlyafter the slippage js attains a value of around 1 ∼ 2 cm. This rapid increase happens as aresult of the dramatic collapse of the snow material. After this, the amount of slip sinkageincreased intermittently with increments in applied load.

From the above series of test, the terrain-track system constants for a representativesnowy terrain can be summarized as having the following values: k2 = 0.315, n2 = 1.220,k4 = 32.34, n4 = 0.862, fs = 1.86 ± 0.35, fm = 0.01, jm = 1 ∼ 2 cm and c0 = 0.685,c1 = 0.694 and c2 = 0.476.

4.4 SUMMARY

In this chapter, we have set the scene for a general study of tracked vehicle systems.Because of the presence of grousers and the existence of the phenomenon of slip-sinkage,the behaviour of tracked running-gear systems is totally different from that of cylindrical

146 Terramechanics

drums and wheels. A new modelling concept is therefore required to predict behaviour. Inthis chapter we have proposed the development of a prediction method based upon a setof experimentally derived parameters. The parameters are developed from small scale,shaped-plate, traction tests. They are scaled up to real machinery sizes using the theory ofdimensional analysis and similitude.

REFERENCES

1. Tamura, Y. & Kaminishi, M. (1980). Improvement of the Tractive Performance with New TrackSystem (Multi-rollers). Komatsu Technical Review, Vol. 26, No. 3, pp. 1–7. (In Japanese).

2. Akai, K. (1986). Soil Mechanics. pp. 86–112. Asakura Press. (In Japanese).3. Janosi, Z. & Hanamoto, B. (1961). The Analytical Determination of Drawbar Pull as a Function

of Slip For Tracked Vehicles. Proc. 1st Int. Conf. on Terrain-Vehicle Systems. Torio.4. Oida, A. (1975). Study on equation of shear stress–displacement curves – on Kacigin-Guskov

equation – J. of Agricultural Machinery, Vol. 37, No. 1, pp. 20–25. (In Japanese).5. Bekker, M.G. (1960). Off-the-road Locomotion. pp. 58–66. The University of Michigan Press.6. Yamaguchi, H. (1984). Soil Mechanics. pp. 141–196. Gihoudou Press. (In Japanese).7. Hata, S. & Hosoi, T. (1981). On the effect of lug pitch upon the tractive effort about track-laying

vehicle, Proc. 7th Int. Conf. ISTVS, Calgary, Canada. Vol. 1, pp. 255–262.8. Muro, T., Omoto, K. & M. Futamura. (1988). Traffic performance of a bulldozer running on

a weak terrain – Vehicle model test, J. of JSCE. No.397/VI-9, pp. 151–157. (In Japanese).9. Ketterer, B. (1981). Modelluntersuchungen zur prognose von schneid und planierkräften im

erdbau. BMT, 7, pp. 355–370.10. Freitag, D.R., Schafer, R.L. & Wismer, R.D. (1970). Similitude Studies of Soil Machine Systems,

Trans. ASAE, Vol. 13, No. 2, pp. 201–213.11. Muro, T. & Kawahara, S. (1986). Size effect of Track Performances of Running Gear on Weak

Ground. J. of JSCE, No. 370/III-5, pp. 105–112. (In Japanese).12. Schafer, R.L., Reaves, C.A. & Young, D.F. (1969). An Interpretation of Distortion in the Simi-

litude of Certain Soil-Machine Systems. Trans. ASAE, Vol. 12, No. 1, pp. 145–149.

EXERCISES

(1) Suppose that three different kinds of standard T shaped steel track of grouser heightH = 3.2 cm and grouser pitch Gp = 5.1, 10.2 and 20.4 cm are standing at rest on asilty loam terrain and that they each are sustaining a contact pressure of p = 9.8 kPa.Calculate the static amount of sinkage s0 for each system by use of the terrain-tracksystem constants data given in Table 4.1. Assume that the value of p0 in Eq. (4.2) is1.96 kPa for all the tracks.

(2) Assume that there are three kinds of standard T shaped steel track equipped withgrousers at a pitch Gp = 5.1, 10.2 and 20.4 cm. Suppose also that the track has a lengthof 70 cm and a width of 20 cm and that the contact pressure on the track is 7.84 kPa.For this arrangement, calculate the tractive resistance T when the track is pulled untila slip value of 5 cm is attained. Use the terrain-track system constants of Table 4.1.

(3) Suppose that a standard T shaped steel track having three kinds of grouser pitch Gp

is slipping under a contact pressure of 7.84 kPa. Calculate the amount of slip sinkagess of the track when the amount of slippage js reaches 7 cm. Assume the terrain-tracksystem constants are as given in Table 4.1.


(4) Imagine that a set of equilateral trapezoidal shaped rubber tracks equipped with fourkinds of grouser of base length L = 2, 3, 4 and 5, as shown in Figure 4.6, are in astationary state on a loose accumulated decomposed granite sandy terrain. The contactpressure is 29.4 kPa. Calculate the static amount of sinkage s0 of each system by useof the terrain-track system constants of Table 4.2.

(5) Equilateral trapezoidal shaped rubber tracks equipped with four kinds of grouser ofbase length L = 2, 3, 4 and 5 are pulled on a loose sandy terrain. The track has a lengthof L of 50 cm and a width of 20 cm and the contact pressure is 34.3 kPa. Calculatethe tractive force T when the amount of slippage reaches 6 cm, through use of theterrain-track system constants of Table 4.2.

(6) Four kinds of equilateral trapezoidal shaped rubber track equipped with grousers havingdifferent base length L are slipping under a contact pressure p = 53.9 kPa. Calculatethe amount of slip sinkage ss when the amount of slippage js reaches 50 cm. Theterrain-track system constants are as shown in Table 4.2.

(7) An equilateral trapezoidal shaped rubber track equipped with a grouser of base lengthL = 4 cm is pulled on a concrete pavement. The track has a length of 70 cm, a widthof 25 cm and is operating under a contact pressure of 29.4 kPa. Calculate the tractiveresistance T when the amount of slippage j reaches 8.5 cm. Assume that the terrain-tracksystem constants are as shown in Table 4.3.

(8) An equilateral trapezoidal shaped rubber track of length 41 cm and width 12.5 cm isoperating in a stationary state on a silty loam terrain. Calculate the static amount ofsinkage for a contact pressure p of 25.5 kPa. How does the amount of sinkage s0 varyfor same contact pressure when the size ratio N increases from 2, to 3 and then to4? Assume that the height of the grouser H is 1.5 cm for a size ratio N = 1, and thatthe coefficients of sinkage k1, k2 and the indices of sinkage n1, n2 are as given inEqs. (4.36) ∼ (4.39).

(9) When the size ratio N of the track given in the previous problem (8) increases throughN = 1, 2, 3 and 4, calculate the variation in the shear resistance τ of the soil at theamount of slippage j = 10 N (cm). Assume that the terrain-track system constants areas given in Eqs. (4.40) ∼ (4.42). When the size ratio N of the track given in the previousproblem (8) increases through N = 1, 2, 3 and 4, calculate the variation in the amountof slip sinkage ss of the tracks at an amount of slippage js = 5N (cm). Assume theterrain-track system constants are as given in Eqs. (4.43) ∼ (4.45).

Chapter 5

Land Locomotion Mechanics for a Rigid-Track Vehicle

5.1 REST STATE ANALYSIS

5.1.1 Bearing capacity of a terrain

When a rigid tracked vehicle of weight W , track belt contact length D, track width B, andamount of eccentricity of center of gravity of vehicle e is in a stationary condition on a hardterrain as shown in Figure 5.1, a ground reaction P acts on the track belt at the positionof the amount of eccentricity e0D and the distribution of the contact pressure takes on theshape of a trapezoid within the bounds |ei| � 1/6. In this case, the amount of sinkage of thetrack belt is negligibly small for a hard terrain. The contact pressures at the front and rearends of the track belt pf and pr can be expressed as follows providing that we assume thatthe hard terrain is a pure linearly-elastic material:

pf = pm(1 − 6e0) (5.1)

pr = pm(1 + 6e0) (5.2)

pm = W

2BD(5.3)

where pm is the average contact pressure. On the other hand, the distribution of the contactpressure becomes a triangle to the rear-side for e0 > l/6 and to the front side for e0 < −1/6.

Figure 5.1. Contact pressure distribution under track belt on hard terrain (at rest).

150 Terramechanics

For these static conditions, the bearing capacity Qc of the terrain can be expressed,following Meyerhof [1], as:

Qc = 2BDe

(cNcqλcq + γBNrqλr

2

)(5.4)

where c is the cohesion, Ncq, Nγq is the coefficient of bearing capacity, λcq, λγ is the shapecoefficient, and De = D(1 − 2e0) is the effective contact length of the track belt whichconsiders the effective loading area. However, when a tractive effort or a braking force actson the vehicle, it is necessary to modify the above equation for inclined load because ofthe inclined and eccentric loads that are applied to the track belt.

When e0 equals zero, Qc can be also expressed following Terzaghi [2] as:

Qc = 2BD

{(1 + 0.3B

D

)cNc +

(0.5 − 0.1B

D

)BγNγ

}(5.5)

where Nc and Nγ are the coefficients of bearing capacity.

5.1.2 Distribution of contact pressures and amounts of sinkage

When a uniform distribution of contact pressure p is applied through a rigid track belt ofwidth B, as shown schematically in Figure 5.2, several equipotential lines of principal stress[3] can be drawn for the region under the track. This can be done through use Boussinesq’selastic solution for a uniformly distributed strip load.

The maximum principal stress σ1 acting at an arbitrary point A on the equipotential lineof the principal stress operates in a direction that bisects the central angle 2ε as shown in thisdiagram. Under the same conditions the minimum principal stress σ3 acts perpendicular toσ1. The values of σ1 and σ3 are thence as follows:

σ1 = (2ε + sin 2ε)p

π(5.6)

σ3 = (2ε − sin 2ε)p

π(5.7)

Figure 5.2. Equi-principal stress lines under track belt.

Rigid-Track Vehicle 151

Further, the horizontal stress σx, the vertical stress σz and the shear stress τxz that acting atthe point A are:

σx = (2ε − sin 2ε · cos 2ψ)p

π(5.8)

σz = (2ε + sin 2ε · cos 2ψ)p

π(5.9)

τxz = sin 2ε · sin 2ψ · p

π(5.10)

Then, the vertical contact pressure σz acting at an arbitrary point in the terrain can becalculated using Eq. (5.9).

On the other hand, the horizontal stress σx, the vertical stress σz and the shear stress τxz

acting at a point B where a vertical line through the edge of the track belt intersects theequipotential line of principal stress can be expressed as:

σx = (2ε − sin 2ε · cos 2ε)p

π(5.11)

σz = (2ε + sin 2ε · cos 2ε)p

π(5.12)

τxz = sin2 2ε · p

π(5.13)

Then, the horizontal stress σx acting on the side part of grouser of the track belt can becalculated by use of Eq. (5.11).

The normal resultant force Fx acting on the side part of grouser over the whole rangeof the contact length of the track belt can be calculated by integrating σx from zero to thegrouser height H as follows:

2ε = cot−1( z

B

)

sin 2ε = B√z2 + B2

, cos 2ε = z√z2 + B2

therefore,

Fx = D∫ H

0σx dz = Dp

π

∫ H

0

{cot−1

( z

B

)− Bz

z2 + B2

}dz

= DpH

π· cot−1

(H

B

)(5.14)

When a rigid-track vehicle is standing stationary on a soft terrain, the rigid track belt willbe generally inclined as a result of an amount of sinkage at the front-idler sf and an amountof sinkage at the rear sprocket sr . The angle of inclination of the vehicle θt0 also variesaccording to the amount of eccentricity e of the center of gravity of the vehicle. As shownin Figure 5.3(a ∼ e), the main part of the track belt BC can take up five different distinctivepositions as a function of the position of the eccentricity e0 relative to that of the groundreaction P. In these cases, the amounts of sinkage sf 0 and sr0 are measured in directionsperpendicular to the normal direction of the main part of the track belt. The distances aretaken from the ground surface to the tip of the grouser.

152 Terramechanics

The amount of sinkage s0(X ) at a distance X from a point B on the front edge of the mainpart of the track belt can be expressed by the following linear function.

s0(X ) = sf 0 + (sr0 − sf 0)X

D(5.15)

where D is the contact length of the rigid track belt.Of necessity, the distribution of the contact pressure under the rigid track belt cannot be

expressed as a linear function. It should be curved in general as shown in the followingequations:

For 0 � s0(X ) � H

p0(X ) = k1{s0(X )}n1 (5.16)

For s0(X ) > H

p0(X ) = p0 + k2{s0(X ) − H }n2 (5.17)

The contact pressure pf 0, the amount of sinkage sf 0 at the contact point B between thefront-idler and the main part of the rigid track belt, the contact pressure pr0 the amount ofsinkage sr0 at the contact point C between the rear sprocket and the main part of the rigidtrack belt can be calculated from the appropriate force and moment equilibrium equations[4] as follows:

(1) For the case where sf 0 ≥ H, sr0 ≥ HUsing the symbols used in Figure 5.3(a), the following equations can be established:

W cos θt0 = 2B∫ D

0p0(X ) dX = 2BDp0 + 2k2BD

n2 + 1× (sr0 − H )n2+1 − (sf 0 − H )n2+1

sr0 − sf 0

(5.18)

Figure 5.3(a). Distribution of contact pressure Pi(X ) and amount of sinkage Si(X ) for sf 0 ≥ H ,sr0 ≥ H .


W cos θt0 ·(

1

2+ e0

)D = 2B

∫ D

0p0(X )X dX

= 2BD2

(sr0 − sf 0)2

[k2

n2 + 2

{(sr0 − H )n2+2 − (sf 0 − H )n2+2}

− k2

n2 + 1(sf 0 − H )

{(sr0 − H )n2+1 − (sf 0 − H )n2+1}

+ 1

2p0(s2

r0 − s2f 0) − p0sf 0(sr0 − sf 0)

](5.19)

From above equations, sf 0 and sr0 can be calculated as follows:

θt0 = tan−1

(sr0 − sf 0

D

)(5.20)

e0 = e + hg + H

Dtan θt0 (5.21)

pf 0 = p0 + k2(sf 0 − H )n2 (5.22)

pr0 = p0 + k2(sr0 − H )n2 (5.23)

(2) For the case where 0 ≤ sf 0 < H < sr0

As shown in Figure 5.3(b), the position X where s0(X ) becomes larger than H for a positiveangle of inclination of vehicle θt0 can be calculated as:

X = H − sf 0

tan θt0= X1

Figure 5.3(b). Distribution of contact pressure Pi(X ) and amount of sinkage Si(X ) for 0 ≤ sf 0 <H < sr0.

154 Terramechanics

thence,

W cos θt0 = 2B∫ X1

0p0(X ) dX + 2B

∫ D

X1

p0(X ) dX

= 2BD

sr0 − sf 0

[k1

n1 + 1(H n1+1 − sf 0

n1+1)

+ (sr0 − H )

{p0 + k2

n2 + 1(sr0 − H )n2

}](5.24)

W cos θt0

(1

2+ e0

)D = 2B

∫ X1

0p0(X )X dX + 2B

∫ D

X1

p0(X ) dX

= 2k1BD2

(sr0 − sf 0)2

{H n1+2 − sf 0

n1+2

n1 + 2− sf 0(H n1+1 − sf 0

n1+1)

n1 + 1

}

+ 2BD2

(sr0 − sf 0)2

{k2

n2 + 2(sr0 − H )n2+2

+ k2

n2 + 1(H − sf 0) (sr0 − H )n2+1

+ 1

2p0(sr0

2 − H 2) − p0sf 0 (sr0 − H )

}(5.25)

From above equations, sf 0 and sr0 can be calculated. Substituting these values intoEqs. (5.20) and (5.21), θt0 and e0 can be determined. For this case we get:

pf 0 = k1(sf 0)n1 (5.26)

pr0 = p0 + k2(sr0 − H )n2 (5.27)

(3) For the case where sf 0 > H > sr0 ≥ 0As shown in Figure 5.3(c), the position X where s0(X ) becomes less than H for a negativeangle of inclination of vehicle θt0 can be calculated as:

X = D + H − sr0

tan θt0= X2

thence,


0p0(X ) dX + 2B

∫ D

X2

p0(X ) dX

= 2k1BD

sr0 − sf 0· sr0

n1+1 − H n1+1

n1 + 1− 2BD

sr0 − sf 0

× (sf 0 − H ){

p0 + k2

n2 + 1(sf 0 − H )n2

}(5.28)


Figure 5.3(c). Distribution of contact pressure Pi(X ) and amount of sinkage Si(X ) forsf 0 > H > sr0 ≥ 0.

W cos θt0·(

1

2+ e0

)D = 2B

∫ X2

0p0(X ) dX + 2B

∫ D

X2

p0(X )X dX

= 2BD2

(sr0 − sf 0)2

{− k2

n2 + 2(sf 0 − H )n2 + 2− k2H

n2 + 1(sf 0 − H )n2+1

+ k2sf 0

n2 + 1(sf 0 − H )n2+1 + 1

2p0(H 2 − sf 0

2) − p0sf 0(H − sf 0)}

+ 2k1BD2

(sr0 − sf 0)2

{1

n1 + 2(sr0

n1+2 − H n1+2)

− sf 0

n1 + 1(sr0

n1+1 − H n1+1)}

(5.29)

From above equations, sf 0 and sr0 can be calculated. Substituting these values intoEqs. (5.20) and (5.21), θt0 and e0 can be determined. The result in this case is

pf 0 = p0 + k2(sf 0 − H )n2 (5.30)

pr0 = k1(sr0)n1 (5.31)

(4) For the case where sf 0 < 0 < H < sr0

As shown in Figure 5.3(d), the position X where s0(X ) becomes larger than H for a positiveangle of inclination of the vehicle θt0 can be calculated as follows:

X = D − L + H

tan θt0= X3

where L is given as follows:

L = sr0

sr0 − sf 0D

156 Terramechanics

Figure 5.3(d). Distribution of contact pressure Pi(X ) and amount of sinkage Si(X ) forsf 0 < H < 0 < sr0.

thence,


D−Lp0(X ) dX + 2B

∫ D

X3

p0(X ) dX

= 2k1BDH n1+1

(n1 + 1)(sr0 − sf 0)+ 2BD(sr0 − H )

sr0 − sf 0×{

p0 + k2

n2 + 1(sf 0 − H )n2

}(5.32)

W cos θt0 ·(

1

2+ e0

)D = 2k1BD2

(sr0 − sf 0)2H n1+1

(H

n1 + 2− sf 0

n1 + 1

)

+ 2BD2

(sr0 − sf 0)2

{1

2p0(sr0

2 − H 2) + k2 (sr0 − H )n2+1

×(

sr0 − H

n2 + 2+ H − sf 0

n2 + 1

)− p0sf 0(sr0 − H )

}(5.33)

From above equations, sf 0 and sr0 can be calculated. Substituting these values intoEqs. (5.20) and (5.21), θt0 and e0 can be determined. In this case,

pf 0 = 0 (5.34)

pr0 = p0 + k2(sr0 − H )n2 (5.35)

(5) For the case where sf 0 > H > 0 > sr0

For the case shown in Figure 5.3(e), the position X where s0(X ) is less than H for a negativeangle of inclination of the vehicle θt0 can be calculated as:

X = LL

(1 − H

sf 0

)= H − sf 0

tan θt0= X4


Figure 5.3(e). Distribution of contact pressure Pi(X ) and amount of sinkage Si(X ) for sf 0 >H > 0 > sr0.

where LL is given as follows:

LL = sf 0

sf 0 − sr0D

thence,


0p0(X ) dX + 2B

∫ LL

X4

p0(X ) dX

= 2BD

sf 0 − sr0

{p0(sf 0 − H ) + k1

n1 + 1H n1+1 + k2

n2 + 1(sf 0 − H )n2+1

}(5.36)

W cos θt0 ·(

1

2+ e0

)D = 2B

∫ X4

0p0(X ) dX + 2B

∫ LL

X4

p0(X ) dX

= 2BD2

(sf 0 − sr0)2

[p0sf 0(sf 0 − H ) + 1

2p0(H 2 + sf 0

2)

+ k2sf 0

n2 + 1(sf 0 − H )n2+1 − k2

{(sf 0 − H )n2+1

n2 + 1+ H (sf 0 − H )n2+1

n2 + 1

}

+ k1

(sf 0

n1 + 1H n1+1 − H n1+2

n1 + 2

)](5.37)

158 Terramechanics

Figure 5.4. Distribution of slip velocity Vs(X ) and amount of slippage j(X ).

From above equations, the values of sf 0 and sr0 can be determined. Substituting these valuesinto Eqs. (5.20) and (5.21), θt0 and e0 can be calculated. For this case the results are:

pf 0 = p0 + k2(sf 0 − H )n2 (5.38)

pr0 = 0 (5.39)

5.2 DRIVING STATE ANALYSIS

5.2.1 Amount of vehicle slippage

When a rigid tracked vehicle is running under traction during driving action on a hard terrainat a constant vehicle speed V and with a circumferential speed of track belt V ′ (V ≤ V ′),the slip velocity Vs of the track plate is constant for the whole range of the contact surfaceof the rigid track belt, as shown in Figure 5.4. However, it is evident that the amount ofslippage j at each of the grouser positions must have a different value.

For a radius of front-idler Rf and of rear sprocket Rr , and given an angular velocity ofthe driving rear sprocket ωr , a slip ratio id for the vehicle can defined as follows:

id = 1 − V

V ′ = Rrωr − V

Rrωr(5.40)

The slip velocity Vs(X ) of the track plate at a distance X from the contact point B betweenthe front-idler and the main part of the rigid track belt has a positive value, and can becalculated as follows:

Vs(X ) = V ′ − V = Rrωr − V (5.41)

The leading grouser � is assumed to move on the terrain only the amount of slippage j0during an elemental time interval �t until the following grouser comes in contact withterrain subsequent to grouser � contacting with terrain, as shown in the figure. During


Figure 5.5. Amount of slippage of vehicle during driving action.

this time, the other grousers �, �, � and � connected to each others will move the sameamount of slippage j0 at the same time.

Therefore, the amount of slippage j of the nth grouser can be computed as the product ofthe proceeding time interval n�t after the beginning of contact and the slip velocity j0/�tas follows:

j = j0�t

· n�t = nj0 (5.42)

Likewise, the amount of slippage j(X ) of the grouser at a distance X can be calculated byintegrating the slip velocity Vs from zero to the required time t = X /Rrωr for the movementof the grouser from the point B to the point X as follows [5]:

j(X ) =∫ t

0(Rrωr − V ) dt = Rrωr − V

Rrωr· X = idX (5.43)

Further, as the track belt moves a distance D around the vehicle itself during driving action,as shown in Figure 5.5, the amount of slippage at the rear end of the track belt BC′ equalsidD while the movement of the vehicle BB′ is (1−id)D, and the required time taken equalsD/V ′.

Thence, the required time td to the point when the vehicle has completely passed by thepoint B can be calculated as:

td = D

V ′ · D

(1 − id)D= 1

1 − id· D

V ′ (5.44)

The amount of slippage js of the point B on the terrain associated with the passage of thevehicle can be calculated as the product of the slip velocity Vs and the proceeding-time ofthe vehicle td as follows:

js = (V ′ − V )td = idV ′ · 1

1 − id· D

V ′ = idD

1 − id(5.45)

5.2.2 Force balance analysis

Figure 5.6 shows the composite of forces that act collectively on a rigid tracked vehiclewhen the vehicle is running under traction during driving action at a slip ratio id on ahorizontal soft terrain. W is the vehicle weight, D is the contact length of the vehicle, Rf

and Rr are the radius of the front-idler and rear sprockets respectively. H is defined as thegrouser height, Gp is the grouser pitch, e is the eccentricity of the center of gravity of thevehicle, hg is the height of the center of gravity G of the vehicle measured from the bottom

160 Terramechanics

Figure 5.6. Forces acting on a rigid tracked vehicle during driving action.

of the track belt, ld is the distance from the center line of the vehicle to the point F ofapplication of the effective tractive effort, hd is the height of the point F measured fromthe bottom of the track belt. V is the vehicle speed, V ′ is the circumferential speed of thetrack belt, and Q is the driving torque acting on the rear sprocket. The parameter sfi is theamount of sinkage of the front-idler measured vertically at the contact point B betweenthe front-idler and the main part of the track belt. Likewise, sri is the amount of sinkageof rear sprocket measured vertically at the contact point C between the rear sprocket andthe main part of the track belt. The parameters pfi and pri are the normal contact pressuresacting at the points B and C of the track belt respectively.

The factor T1 is the driving force acting on the circumferential lower part of the rearsprocket and T1 = Q/Rr . T2 is the compaction resistance which acts horizontally at a depthzi on the front part of the track belt. T3 is the thrust developed along the rigid track beltthrough each grouser as the sum of the shear resistances acting on the slip lines connectingeach grouser tip. T4 is the effective driving force which acts horizontally through the point F.P is the ground reaction acting normally to the contact part of the track belt. The eccentricityof P is ei. The angle θti is the angle of inclination of the vehicle. This angle can be calculatedfrom the amounts of sinkage sfi and sri.

The static amount of sinkage s0(X ), taken as being normal to the main part of the rigidtrack belt at a distance X from the point B, can be expressed as a linear function of X usingthe static amount of sinkage sf 0i and sr0i as follows:

s0i(X ) = sf 0i + (sr0i − sf 0i)X

D(5.46)

Thence, the distribution of the contact pressure pi(X ) should be curved and it should takethe following form:

For 0 � s0(X ) � H

pi(X ) = k1{s0i(X )}n1 (5.47)


For s0(X ) > H

pi(X ) = k1H n1 + k2{s0i(X ) − H }n2 (5.48)

The balance of forces acting on the vehicle in the horizontal and vertical directions and themoment balance around the rear axle can be expressed by the following equations:

T2 + T4 = T3 cos θti − P sin θti (5.49)

W = T3 sin θti + P cos θti (5.50)

Q = T1Rr = T3Rr (5.51)

θti = sin−1(

sri − sfi

D

)(5.52)

Then, the effective driving force T4 and the ground reaction P can be calculated as:

T4 = T3

cos θti− W tan θti − T2 (5.53)

P = W

cos θti− T3 tan θti (5.54)

For id = 0, the above equation will be satisfied if T2 = T4 = 0 and T3 = W sin θt0, θti = θt0

and P = W cos θt0.Next, for the pure rolling state of the vehicle, it is proposed that T1 = T3 = W sin θti,

T4 = −T2 and P = W cos θti, and that the depth zi where the compaction resistance T2 actscan be calculated using the rut depth of the vehicle e.g. the amount of sinkage of rearsprocket sri and the angle of inclination of vehicle θti, as follows:

Zi = Sri −[

T2

{Rr + (hd − Rr) cos θti −

(Id − D

2

)× sin θti

}

−W

{hg sin θti − D

(1

2− e

)cos θti

}]/(T2 + W cos θti

tan θti

)(5.55)

We assume here that the moment around the axle of the rear sprocket due to T2, T3, P andW are zero or are negligibly small.

Additionally, the moment balance equation operating as a result of a composite of forcesacting on the rigid tracked vehicle during driving action with slip ratio id can be expressed as:

−Q + T3Rr − T2(Rr − sri + zi) + T4

{(hd − Rr) cos θti −

(ld − D

2

)sin θti

}

+ W

{(hg − Rr) sin θti − D

(1

2− e

)cos θti

}+ PD

(1

2− ei

)= 0 (5.56)

162 Terramechanics

Substituting the previous Eq. (5.51) into the above equation, the expression can be solvedfor eccentricity ei as follows:

ei = 1

2+ 1

PD

[−T2(Rr − sri + zi) + T4

{(hd − Rr) cos θti −

(ld − D

2

)sin θti

}

+ W

{(hg − Rr) sin θti − D

(1

2− e

)cos θti

}](5.57)

For id = 0, P = W cos θt0 and T2 = T4 = 0, when a rigid tracked vehicle is stopped underconditions of T3 = W sin θt0 and Q = WRr sin θt0. In this case, the eccentricity e0 of theground reaction P can be calculated as:

e0 = e + hg − Rr

Dtan θt0 (5.58)

This notion becomes important when the vehicle is climbing up a slope.It is noted here that the value of e0 given in Eq. (5.21) is different from the above

mentioned value because the rigid track belt at rest was calculated for Q = 0 assuming thatthe track belt is a rotational rigid body supported by the bearing capacity of terrain at thefront and rear contact part or by the braking action.

5.2.3 Thrust analysis

As shown in Figure 5.7, the thrust T3 developed on the contact part of a track belt [6] iscomprised of the sum Tmb of the shear resistances τm(X ) acting on the base area of grousersof the main part of track belt, and the sum Tms of the shear resistances τms(X ) acting onthe side parts of the grousers. Additionally there are components representing the sum Tfb

Figure 5.7. Stresses acting on main part of track belt, parts of front and rear wheel and distributionof amount of slippage (during driving action).


of the shear resistances τf (θ) acting on the base area of grousers of the contact part ofthe front-idler, the sum Tfs of the shear resistances τfs (θ) acting on the side parts of thegrousers, and the sum Trb of the shear resistances τr(δ) acting on the base area of grousersof the contact part of the rear sprocket. These forces are supplemented by the summationTrs of shear resistances τrs(δ) acting on the side parts of the grousers, which develop withincreasing amounts of sinkage. Overall the total picture is as follows;

T3 = Tmb + Tms + Tfb + Tfs + Trb + Trs (5.59)

where X is the distance from the point B on the main part of the track belt, θ is the centralangle ∠BOM for an arbitrary contact point M on the front-idler, and δ is the central angle∠DO′N between the end point D and an arbitrary point N on the contact part of the rearsprocket.

(1) Main part of track belt [7]The amount of static sinkage s0i(X ) of the main part of track belt BC and the distribution ofthe contact pressure pi(X ) can be calculated from Eqs. (5.46) and (5.47), (5.48) respectively.Since the main part of the track belt is inclined at a vehicle inclination angle of θti duringdriving action, the slip ratio i′d needs to be modified using id given in Eq. (5.40). This maybe done as follows:

i′d = 1 − V

cos θti· 1

V ′ = 1 − 1 − idcos θti

(5.60)

Thence, the distribution of the amount of slippage j(X ) is given for an amount of slippagejB at the point B as:

j(X ) = jB + i′dX (5.61)

The distribution of shear resistance τm(X ) acting on the base area of the grousers of themain part of track belt can be calculated from the previous Eq. (4.5) as follows:

τm(X ) = {mc + mf · pi(X )

} [1 − exp

{−a( jB + i′dX )}]

(5.62)

Then, the overall thrust Tmb acting on the base area of grousers of the main part of the trackbelt can be calculated as:

Tmb = 2B∫ D

0τm(X ) dX (5.63)

Further, the distribution of the shear resistance τms(X ) that acts on the four sides of thegrousers of the main part of track belt can be calculated from the average normal stressFx/DH acting on the side part of the track belt given in Eq. (5.14) as:

tms(X ) = 4H

{mc + mf · pi(X )

πcot−1

(H

B

)}× [

1 − exp{−a( jB + i′dX )

}](5.64)

Then, the thrust Tms acting on the side part of grousers of the main track belt can bedetermined as:

Tms =∫ D

0tms(X ) dX (5.65)

164 Terramechanics

Since the above calculations correspond with sf 0 = sf 0i, sr0 = sr0i in the cases of the Section5.1.2 (1), (2), (3), pi(X ) can be calculated for each amount of sinkage s0i(X ).

Now, let us consider the situation where sf 0i < 0 < H < sr0i as in Section 5.1.2(4). Sub-stituting W cos θt0 = P, p0(X ) = pi(X ) and ei = e0 into Eqs. (5.32), (5.33), the amount ofsinkage sf 0 = sf 0i and sr0 = sr0i can be calculated. The real contact length L of the main partof the track belt is thence calculated as:

L = sr0i

sr0i − sf 0iD (5.66)

For 0 � X � D − L the contact pressure pi(X ) becomes zero and then the shear resistanceτm(X ) becomes zero. For D − L < X � D, pi(X ) can be calculated from Eqs. (5.47), (5.48)by use of the amount of sinkage s0i(X ) given in Eq. (5.46) and then τm(X ) can be determinedfrom the following expression.

τm(X ) = {mc + mf · pi(X )

} [1 − exp

{−ai′d(X − D + L)}]

(5.67)

Then, the thrust Tmb acting on the base area of the grousers of the main contact part of thetrack belt can be calculated as follows:

Tmb = 2B∫ D

D−Lτm(X ) dX (5.68)

Following this, the shear resistance τms(X ) acting on the four sides of the grousers of themain contact part of the track belt may be calculated as:

tms(X ) = 4H

{mc + mf · pi(X )

πcot−1

(H

B

)}× [

1 − exp{−ai′d(X − D + L)

}](5.69)

Next, the thrust Tms acting on the side of grousers of the main contact part of the track beltcan be calculated as follows:

Tms =∫ D

D−Ltms(X ) dX (5.70)

Further, let us consider the case where sf 0i > H > sr0i as in Section 5.1.2(5). Calculating theamount of sinkage sf 0i and sr0i from Eqs. (5.36), (5.37) in the same manner as previously,the real contact length LL of the main part of the track belt can be calculated as follows:

LL = sf 0i

sf 0i − sr0iD (5.71)

For LL < X � D, the contact pressure pi(X ) becomes zero and then the shear resistanceτm(X ) becomes zero. For 0 � X � LL, pi(X ) can be calculated from Eqs. (5.47), (5.48) byuse of the amount of sinkage s0i(X ) given in Eq. (5.46) and then τm(X ) can be given asfollows:

τm(X ) = {mc + mf · pi (X )

} [1 − exp

{−a( jB + i′dX )}]

(5.72)


Then, the thrust Tmb acting on the base area of grousers of the main contact part of the trackbelt can be calculated as follows:

Tmb = 2B∫ LL

0τm(X ) dX (5.73)

Moreover, the shear resistance tms(X ) acting on the four sides of grousers of the maincontact part of the track belt is given as:

tms(X ) = 4H

{mc + mf · pi(X )

πcot−1

(H

B

)} [1 − exp

{−a(jB + i′dX )}]

(5.74)

thence, the thrust Tms acting on the side parts of the grousers of the main contact part ofthe track belt can be calculated as:

Tms =∫ LL

0tms(X ) dX (5.75)

(2) Contact part of front-idlerAs the front-idler comes into contact with the terrain at point A (as shown in Figure 5.7),the entry angle θf = ∠AOB can be calculated as follows:

θf = cos−1

(cos θti − sfi

Rf + H

)− θti (5.76)

The amount of slippage jf (θ) at the contact part of the front-idler can be calculated throughthe use of the entry angle θf 0 and the angle of inclination of vehicle θt0i (which in turn iscalculated from the static amounts of sinkage sf 0i and sr0i). Using these ideas we get:

jf (θ) = (Rf + H )∫ θf 0

θ

{1 − (1 − id) cos(θ + θt0i)

}dθ

= (Rf + H )[(θf 0 − θ) − (1 − id)

{sin(θf 0 + θt0i) − sin(θ + θt0i)

}](5.77)

θf 0 = cos−1

(cos θt0i − sf 0i cos θt0i

Rf + H

)− θt0i (5.78)

θt0i = tan−1

(sr0i − sf 0i

D

)(5.79)

In addition, the amount of slippage jB at the point B is given as jf (0) from the aboveEq. (5.77).

The amount of sinkage sf 0i(θ) at the contact part of the front-idler is given as follows:

sf 0i(θ) = (Rf + H ){cos(θ + θt0i) − cos(θf 0θt0i)

}cos(θ + θt0i)

(5.80)

then the contact pressure pfi(θ) can be calculated as in the following equations:

For 0 � sf 0i(θ) � H

pfi(θ) = k1{sf 0i(θ)

}n1 (5.81)

166 Terramechanics

For sf 0i(θ) > H

pfi(θ) = k1H n1 + k2{sf 0i(θ) − H

}n2 (5.82)

Then, the shear resistance τf (θ) acting on the contact part of the front-idler can be calculatedas follows:

τfi(θ) = {mc + mf · pfi(θ)

} [1 − exp

{−ajf (θ)}]

(5.83)

Thence, the thrust Tfb acting on the base area of grousers of the front-idler in the directionof the main part of the track belt BC can be determined as follows:

Tfb = 2B(Rfg + H )∫ θf 0

0

{τf (θ) cos θ − pfi(θ) sin θ

}dθ (5.84)

Further, the shear resistance tfs(θ) acting on the four sides of the grousers on the front-idleris given as:

tfs(θ) = 4H

{mc + mf · pfi(θ)

πcot−1

(H

B

)} [1 − exp

{−ajf (θ)}]

(5.85)

then the thrust Tfs acting on the side part of the front-idler can be calculated:

Tfs = (Rf + H )∫ θf 0

0tfs(θ) cos θdθ (5.86)

The above calculations correspond to the cases of Section 5.1.2(1), (2), (3) and (5). For thecase of Section 5.1.2 (4), both the values of Tfb and Tfs become zero.

(3) Contact part of rear sprocketAs the rear sprocket separates finally from the terrain at a point D as shown in Figure 5.7,the exit angle θr = ∠CO′D is almost equal to the angle of inclination of the vehicle θti

providing that we assume that the expansive amount of deformation of the terrain due tounloading is negligibly small.

The amount of slippage jr(δ) at the contact part of the rear sprocket can be calculatedthrough use of the exit angle θr0 = θt0i and the angle of inclination of the vehicle θt0i whichin turn is calculated from the static amount of sinkage of the front and rear wheel sf 0i andsr0i. The calculation equation is as follows:

jr(δ) = (Rr + H ) {(θt0i − δ) − (1 − id)(sin θt0i − sin δ)} + i′dD + jB (5.87)

Additionally, the amount of slippage jD at the point D is given as jr(0) from the aboveequation.

The amount of sinkage sr0i(δ) at the contact part of the rear sprocket is given as follows:

sr0i(δ) = [(Rr + H )

{cos θt0i − cos (θf 0 + θt0i)

}+ D sin θt0i

+ (Rr + H )(cos δ − cos θt0i)] / cos δ (5.88)


Thence the contact pressure pri(δ) can be calculated as follows:

For 0 � sr0i(δ) � H

pri(δ) = k1{sr0i(δ)}n1 (5.89)

For sr0i(δ) > H

pri(δ) = k1H n1 + k2{sr0i(δ) − H }n2 (5.90)

Following this, the shear resistance τr(δ) acting on the contact part of the rear sprocket canbe determined by the following expression.

τr(δ) = {mc + mf · pri(δ)

}[1 − exp{−ajr(δ)}] (5.91)

Thence, the thrust Trb acting on the base area of the grousers of the rear sprocket in thedirection of the main part of the track belt BC can be calculated as:

Trb = 2B(Rr + H )∫ θr0

0{τr(δ) cos(θt0i − δ) + pri(δ) sin(θt0i − δ)} dδ (5.92)

Additionally, the shear resistance trs(δ) acting on the four sides of the grousers of the rearsprocket is given as:

trs(δ) = 4H

{mc + mf · pri(δ)

πcot−1

(H

B

)}[1 − exp{−ajr(δ)}] (5.93)

Then, the thrust Trs acting on the side parts of the grousers of the rear sprocket can becalculated as:

Trs = (Rr + H )∫ θr0

0trs(δ) cos (θt0i − δ) dδ (5.94)

The above calculations correspond to the cases of Sections 5.1.2 (1), (2), (3) and (5). In thecase of Section 5.1.2 (4), the values of Trb and Trs can be determined by substituting thefollowing expressions into Eqs. (5.91), (5.92) and (5.93), (5.94), respectively.

jr(δ) = (Rr + H ){(θt0i − δ) − (1 − id)(sin θt0i − sin δ)} + i′dL (5.95)

sr0i(δ) = L sin θt0i + (Rr + H )(cos δ − cos θt0i)

cos δ(5.96)

For the above calculations of the thrust of a rigid tracked vehicle running on a flat terrain,the constant amount of slippage jw which occurs on the whole range of contact part of thetrack belt due to the force W sin θti is assumed to have a negligibly small value.


When a rigid tracked vehicle is running on a horizontal soft terrain, it makes a rut of depthsri which corresponds to the amount of sinkage of the rear sprocket. To calculate the depthof this rut, the amount of sinkage of the rigid tracked vehicle has to be considered duringthe driving state.

168 Terramechanics

The total amount of sinkage sfi and sri of a rigid track belt can be calculated from thestatic amounts of sinkage sf 0i and sr0i and the amounts of slip sinkage sfs and srs of frontand rear wheel (which are normal to the main part of the track belt) as follows:

sfi = (sf 0i + sfs) cos θti (5.97)

sri = (sr0i + srs) cos θti (5.98)

where sf 0i and sr0i equal s0i(0) and s0i(D) – respectively calculated from Eq. (5.46) – andθti is given by Eq. (5.52).

The amount of slip sinkage sfs at the point B on the front-idler can be calculated bysubstituting the contact pressure pfi(θm) and the amount of slippage of the soil jfs into theprevious Eq. (4.8). This yields the expression:

sfs = c0

M∑m=1

{pfi(θm) cos θm

}c1

{(m

Mjfs

)c2

−(

m − 1

Mjfs

)c2}

(5.99)

where

θm = θf 0

(1 − m

M

)The parameter θf 0 can be calculated from Eq. (5.78) and pfi(θm) can be calculated fromEqs. (5.81) and (5.82). The amount of slippage of soil jfs of the contact part of the front-idlercan be determined as:

jfs = (Rr + H ) sin θf · id1 − id

(5.100)

Following this, the amount of slip sinkage srs at the point C on the rear sprocket can becalculated as follows [8] by substituting the contact pressure pi(X ) and the amount ofslippage of soil js into the previous Eq. (4.8). Doing this, we get:

srs = sfs + c0

N∑n=1

{pi

(nD

N

)}c1[(

njsN

)c2

−{

(n − 1) jsN

}c2]

(5.101)

where

js = i′dD

1 − i′d

Furthermore, the total amount of sinkage si(X ) of the rigid track belt needs to satisfy thefollowing equation.

si(X ) = sfi + (sri − sfi)X

D(5.102)

The compaction resistance T2 can be calculated from the total amount of sinkage sri of therear sprocket obtained from above equations. That is, as shown in Figure 5.8, when a rigidtracked vehicle of track width B and contact length D moves through a distance Lx, thework T2Lx to make the rut of ABDD′A′ can be assumed to be equal the work necessary to


Figure 5.8. Work due to compaction resistance T2.

compact the terrain of the area of 2BLX corresponding to the contact part of track belt tothe depth sri as follows:

T2 · Lx = 2B · Lx

∫ sri

0p ds0

Now, using the previous Eqs. (4.1) and (4.2), the compaction resistance T2 can becalculated as:

T2 = 2B

[∫ H

0k1s0

n1 ds0 +∫ sri

H

{k1H n1 + k2(s0 − H )n2

}ds0

]

= 2k1B

n1 + 1H n1+1 + 2k1BH n1 (sri − H ) + 2k2B

n2 + 1(sri − H )n2+1 (5.103)

The above calculations can be applied in the case of 0 � sf 0i � sr0i.In the situation where sf 0i > sr0i > 0, the total amount of sinkage of the rear sprocket

following the passage of the front-idler should, of necessity, be larger than the total amountof sinkage of the front-idler. Therefore, Eq. (5.98) can be rewritten as follows:

sri = (sf 0i + sr0i) cos θti = sfi + (srs − sfs) cos θti (5.104)

Moreover, for the situation where sf 0i < 0 < H < sr0i, sfs = 0 and sfs can be calculated, byuse of the factor L given in Eq. (5.66), as follows:

srs = c0

N∑n=1

{pi

(nL

N

)}c1[(

njsN

)c2

−{

(n − 1)jsN

}c2]

(5.105)

where

js = i′dL

1 − i′dThence, the total amount of sinkage sfi and sri can be calculated by substituting the valuesof sfs and srs obtained from the above equation into Eqs. (5.97), (5.98), respectively.

Next, for the situation where sf 0i > H > 0 > sr0i, sfs and sfi can be calculated respectivelyfrom Eqs. (5.99), (5.97). The parameter srs can be calculated, by use of the parameter LLgiven in Eq. (5.71), as follows:

srs = sfs + c0

N∑n=1

{pi

(nLL

N

)}c1[(

njsN

)c1

−{

(n − 1)jsN

}c2]

(5.106)

170 Terramechanics

where

js = i′dLL

1 − i′d

After this, the total amount of sinkage of rear sprocket sri following the passage of thefront-idler can be calculated by use of Eq. (5.104).

For each of the situations mentioned above, the compaction resistance T2 can be cal-culated by substituting the obtained total amount of sinkage of the rear sprocket sri intoEq. (5.103).

5.2.5 Energy equilibrium equation

The effective input energy E1 supplied by a driving torque applied to the rear sprocket mustequals the total output energy i.e. the sum of the compaction energy E2 required to makea rut under the track belt, the slippage energy E3 that occurs during shear deformation ofthe soil under the track belt plus the effective driving force energy E4 which is the workdone by the drawbar pull of the rigid tracked vehicle. Thus the following energy equilibriumequation can be set up.

E1 = E2 + E3 + E4 (5.107)

Next, let us consider the situation where a track belt moves a distance of contact length Dof in relation to the rigid tracked vehicle. As the vehicle moves horizontally the distanceof(1 − i′d

)D cos θti during this time, the various energy components can be calculated as

follows:

E1 = T1D = T3D (5.108)

E2 = T2(1 − i′d)D cos θti = T2(1 − id)D (5.109)

E3 = T3i′dD + W (1 − i′d)D sin θti = T3

(1 − 1 − id

cos θti

)D + W (1 + id)D tan θti (5.110)

E4 = T4(1 − i′d)D cos θti = T4(1 − id)D (5.111)

Further, each energy component per unit of time can be calculated using the vehicle speedV and the circumferential speed of track belt V ′ as follows:

E1 = T1V ′ = T3V ′ = T3V

1 − id(5.112)

E2 = T2(1 − id)V ′ = T2V (5.113)

E3 = T3

(1 − 1 − id

cos θti

)V ′ + W (1 − id)V ′ tan θti

= T3

(1

1 − id− 1

cos θti

)V + WV tan θti (5.114)

E4 = T4(1 − id)V ′ = T4V (5.115)


5.2.6 Effective driving force

The effective driving force T4 that a rigid tracked vehicle develops during driving action canbe calculated by substituting the thrust T3, the compaction resistance T2 and the angle ofinclination of the vehicle θti into Eq. (5.53). In the relationship between the effective drivingforce T4 and slip ratio id , an ‘optimum slip ratio’ idopt is defined as the slip ratio at which theeffective driving force energy E4 takes on a maximum value for a constant circumferentialspeed of track belt V ′. Likewise an ‘optimum effective driving force’ Tdopt is defined asthe effective driving force corresponding to the optimum slip ratio. Additionally, a ‘tractivepower efficiency’ factor Ed can be defined as:

Ed = E4

E1= VT4

V ′T1= (1 − id)

T4

T1(5.116)

Figure 5.9 presents a flow chart that may be used to calculate the tractive performance of arigid tracked vehicle running on a soft terrain during driving action. Initially, the geometricaldimensions of the vehicle W , B, D, Rf , Rr , H , e, hg, ld , hd and V ′ are required as inputdata. Given these, the terrain-track system constants k1, n1, k2, n2, mc, mf , a, c0, c1, andc2 can be read. After this, the contact pressure distributions pf 0, pr0 and p0(X ) of the rigidtracked vehicle at rest can be calculated for the static amount of sinkage distributions sf 0,sr0 and s0(X ) and the angle of inclination of the vehicle θt0 for each of the cases (1) ∼ (5) asmentioned in Section 5.1.2. This factor should be calculated iteratively until the magnitudeof the sinkage s0(X ) is strictly determined. Next, the slip ratio id during driving action canbe taken from id = 1% to 99% successively.

The calculation flow system can be divided into the following three streams or strandsaccording to the static amount of sinkage of the front-idler sf 0i and the rear sprocket sr0i.

For the situation where sf 0i ≥ H , sr0i ≥ H ; 0 ≤ sf 0i < H < sr0i; sf 0i > H > sr0i � 0, s0i(X )and θtio can be calculated from Eqs. (5.46) and (5.47) respectively. Then pfi, pri and pi(X )can be calculated using Eqs. (5.47) and (5.48). After this, the amounts of slip sinkage sfs andsrs can be calculated from Eqs. (5.99) and (5.101) respectively. Thence, the total amountof sinkage of the front-idler sfi and the rear sprocket sri can be calculated from Eqs. (5.97)and (5.98) respectively in the case of 0 � sf 0i � sr0i, and from Eqs. (5.97) and (5.104)respectively in the case of sf 0i > sr0i > 0. Following this, s0(X ) and θti can be determinedby use of Eqs. (5.102) and (5.52), respectively.

In a next step, the compaction resistance T2 can be calculated by substituting sri intoEq. (5.103). The thrust T3 can then be calculated from Eq. (5.59) as the sum of the factorTmb given in Eq. (5.63), Tms given in Eq. (5.65), Tfb given in Eq. (5.84), Tfs given inEq. (5.86), Trb given in Eq. (5.92) and Trs given in Eq. (5.94). Following this, the drivingforce T1 given in Eq. (5.51), the effective driving force T4 given in Eq. (5.53) and the groundreaction P given in Eq. (5.54) can be calculated by iteration until the thrust T3 is uniquelydefined.

It should be noted that in the process of calculation of the static amounts of sinkage ofthe front-idler and rear sprockets, the ground reaction P′ acting on the main part of the rigidtrack belt should be calculated by subtracting the following ground reaction Pf acting onthe contact part of the front-idler from the above mentioned ground reaction P.

Pf = 2B(Rf + H )∫ θf 0

0

{τf (θ) sin θ + pfi(θ) cos θ

}dθ (5.117)

172 Terramechanics

Figure 5.9. Flow chart to calculate the tractive performance of a rigid tracked vehicle running on softterrain during driving action.

where τf (θ) and pfi(θ) may be calculated from Eqs. (5.83) and (5.81), (5.82) respectivelyand θf 0 can be calculated from Eq. (5.78).

In addition, in the cases (1), (2), (3) given in the Section 5.1.2, sf 0i and sr0i may becalculated by substituting the value of P′ = P − Pf into W cos θt0.


For cases where sf 0i < 0 < H < sr0i, s0i(X ) and θt0i and pfi (=0), pri, pi(X ) can be calcu-lated in a similar manner. The amount of slip sinkage of the rear sprocket sri can be calculatedfrom Eq. (5.105), while that of the front-idler sfs is zero. Thence, the total amount of sinkageof the front-idler sfi and the rear sprocket sri can be calculated from Eqs. (5.97) and (5.98)respectively and then si(X ) and θti can be determined.

Following this, the compaction resistance T2 can be calculated from the obtained amountof sinkage sri. The thrust T3 can be calculated as the sum of the elements Tmb given inEq. (5.68), Tms given in Eq. (5.70), Trb given in Eq. (5.92) and Trs given in Eq. (5.94). Thedriving force T1, the effective driving force T4 and the ground reaction P can be calculatediteratively until the thrust T3 is uniquely defined.

For the case where sf 0i > H > 0 > sr0i, s0i(X ) and θt0i and pfi, pri (=0), pi(X ) can becalculated in the same way. The amount of slip sinkage of the front-idler sfs can be calculatedfrom Eq. (5.99) and the amount of slip sinkage of the rear sprocket srs can be calculatedfrom Eq. (5.106). Thence, the total amount of sinkage of the front-idler sfi and the rearsprocket sri can be calculated from Eqs. (5.97) and (5.104) respectively and then si(X ) andθti can be determined.

Next, the compaction resistance T2 can be calculated from the resultant amount ofsinkage sri.

In the calculation of the thrust T3, the value of the factors Tmb given in Eq. (5.68) and theTms given in Eq. (5.70) as the sum of the shear resistance of the soil acting on the base andareas of grousers of the main part of the track belt, and the factors Trb given in Eq. (5.92)and Trs given in Eq. (5.94) as the sum of soil acting on the base and side areas of grousers ofthe contact part of the rear sprocket are required to be modified. This modification derivesfrom considering the fact that that the rear part of the track belt that follows the passage ofthe front-idler comes into contact with the terrain. Then the contact part which lies in therange of LL < X < D develops some cohesive resistance – given as a function of mc – evenif the contact pressure pi(X ) becomes zero. The modified values are then as follows:

Tmb = 2B

∥∥∥∥∫ LL

0τm(X ) dX +

∫ D

LLmc[1 − exp{−a( jB + i′dX )} dX ]

∥∥∥∥ (5.118)

Tms =∫ LL

0tms(X ) dX + 4H

∫ D

LLmc[1 − exp{−a( jB + i′dX )}] dX (5.119)


0mc[1 − exp{−ajr(δ)}] dδ (5.120)

Trs = 4H (Rr + H )∫ θr0

0mc[1 − exp{−ajr(δ)}] cos (θt0i − δ) dδ (5.121)

Then, the thrust T3 can be determined as the sum of the above Tmb, Tms, Trb, Trs plus thecomponents Tfb given in Eq. (5.84), and Tfs given in Eq. (5.86).

By use of the finally values obtained from the above calculations of T1, T2, T3, T4, P, sfi,sri and θti, each of the energy components given in Eqs. (5.108)–(5.115) and the tractiveefficiency Ed given in Eq. (5.116) can be determined. Also, the distributions of the amountof slippage j(X ), the contact pressure distribution pi(X ), the shear resistance τi(X ) underthe track belt and the distribution of the amount of sinkage of the track belt si(X ) can becalculated. The above calculation can be recursively carried out for various slip ratios id ,

174 Terramechanics

Figure 5.10. Distribution of slip velocity Vs(X ) and amount of slippage j(X ) under track belt (duringbraking action).

so that T1 ∼ T4 − id curves, sfi, sri − id curves, ei, θti − id curves, E1 ∼ E4 − id curves andEd − id curve can be drawn graphically. At the same time, the optimum effective drivingforce T4opt , the optimum slip ratio idopt , the maximum effective driving force Tm and thecorresponding slip ratio im can be determined.

5.3 BRAKING STATE ANALYSIS

5.3.1 Amount of vehicle slippage

When a rigid tracked vehicle is running during braking action on a hard terrain at a constantvehicle speed V and a constant circumferential speed of track belt V ′ (V > V ′), the slipvelocity Vs of a track plate is constant for the whole range of the contact part of the rigidtrack belt, as shown in Figure 5.10, but the amount of slippage j at each grouser positionhas a different value.

For a radius of rear sprocket Rr and angular velocity of the braking rear sprocket ωr , theskid ib of the vehicle can be defined as:

ib = V ′

V− 1 = Rrωr − V

V(5.122)

The slip velocity Vs(X ) of the track plate at some distance X from the contact point Bbetween the front-idler and the main part of the rigid track belt takes on a negative value.It can be calculated from:

Vs(X ) = V ′ − V = Rrωr − V (5.123)

Again, the amount of slippage j(X ) of a grouser at a distance X can be calculated byintegrating the slip velocity Vs from zero to the required time t = X /Rrωr for the movement


Figure 5.11. Amount of slippage of vehicle during braking action.

of the grouser from the point B to the point X. The calculation is as follows and it hasa negative value.

j(X ) =∫ t

0(Rrωr − V ) dt = Rrωr − V

RrωR· X = ib

1 + ib· X (5.124)

Yet again, when the track belt moves a distance D around the vehicle itself during brakingaction, as shown in Figure 5.11, the amount of slippage at the rear end of the track belt BC′equals −ibD/(1 + ib) while the movement of the vehicle BB′ becomes D/(1 + ib), and therequired time is given as D/V ′ = D/{V (1 + ib)}.

Thence, the required time tb when the vehicle has passed totally past the point B can becalculated as:

tb = D

V= (1 + ib)

D

V ′ (5.125)

The amount of slippage js of the soil at the point B in the terrain, associated with the passageof the vehicle, can be calculated as the product of the slip velocity Vs and the transit timeof the vehicle tb as follows:

js = (V ′ − V )tb = ibV · (1 + ib)D

V ′ = ibD (5.126)

5.3.2 Force balance analysis

Figure 5.12 shows the composite of forces that act on a rigid tracked vehicle when the vehicleis running during braking action at a skid ib on a horizontal soft terrain. The symbols W,D, B, Rf , Rr , H , Gp, e, hg , ld , hd , V , V ′, sfi, sri and pfi, pri have already been explained inSection 5.2.2.

T1 is the braking force acting tangentially to the upper part of the rear sprocket, and isequivalent to the braking torque Q divided by the radius of the rear sprocket Rr . T2 is thecompaction resistance that acts horizontally on the front part of the track belt. Its point ofapplication lies at a depth zi given in Eq. (5.55). T3 is the drag developed along the rigidtrack belt through each grouser as the sum of the shear resistance acting on the slip linesthat connect each grouser tip. T4 is the effective braking force that acts horizontally throughthe point F. P is the ground reaction acting normally on the contact part of the track belt.The eccentricity ei of P is given in Eqs. (5.57) and (5.58). θt is the angle of inclination ofthe vehicle. It can be calculated from sfi and sri. The static amount of sinkage s0i(X ) of themain part of the track belt can be calculated from Eq. (5.46). The distribution of contactpressure pi(X ) under the track belt can be calculated from Eqs. (5.47) and (5.47).

176 Terramechanics

Figure 5.12. Forces acting on a rigid tracked vehicle during braking action.

The equations for force equilibrium in the horizontal and vertical directions and formoment balance around the rear axle have already been established, for driving action,as Eqs. (5.49) ∼ (5.51) and (5.56). During braking action, all the values of Q, T1, T3 andT4 should be calculated as having negative values.

Thence, the braking force T1 and the effective braking force T4 can be expressed as:

T1 = T3 (5.127)

T4 = T3

cos θti− W tan θti − T2 (5.128)

In the above equations, the method of calculation of the drag T3 and the compactionresistance T2 will be as outlined in the following two Sections.

5.3.3 Drag

As shown in Figure 5.13 the drag T3 developed on the contact part of the track belt canbe expressed as the sum of a number of components, namely: Tmb which is the integral ofthe shear resistance τm(X ); the component Tms which is the integral of the shear resistancetms(X ) which acts on the base and side areas of grousers of the main part of track belt;the component Tfb which is the integral of the shear resistance τf (θ); the component Tfs

which is the integral of the shear resistance tfs(θ) which acts on the base and side areas ofgrousers of the contact part of the front-idler respectively; the component Trb which is theintegral of the shear resistance τr(δ) and the component Trs which is the integral of the shearresistance trs(δ) which acts on the base and side areas of grousers of the contact part of therear sprocket respectively. These shear resistances occur with increases in the amount ofsinkage of the track belt. Consequently, the drag T3 can be expressed as the sum of Tmb,Tms, Tfb, Tfs, Trb and Trs as shown in Eq. (5.59).

(1) Main part of track beltAs a consequence of the main part of the track belt being sloped at an angle equal to thatof the angle of inclination of the rigid tracked vehicle θti during braking action, the skid i′b


Figure 5.13. Stresses acting on main part of track belt, parts of front and rear wheel and distributionof amount of slippage (during braking action).

needs to be modified – using the value of ib given in Eq. (5.122) – as follows:

i′b = V ′

Vcos θti − 1 = (1 + ib) cos θti − 1 (5.129)

Then, the distribution of amount of slippage j(X ) on the main part of the rigid track beltcan be given for an amount of slippage jB at point B as:

j(X ) = jB + i′b1 + i′b

X (5.130)

When jB equals the maximum amount of slippage jf max on the contact part of the front-idler,the distribution of shear resistance τm(X ) acting on the base area of the grousers of the mainpart of the track belt can be calculated from the previous Eq. (4.5) as follows:

For the untraction state of jq < j(X ) < jp = jB = jf max

τ ′m(X ) = τp − k0

{jp − j(X )

}n0 (5.131)

For the reciprocal traction state of j(X ) � jq

τ ′′m(X ) = −{mc + mf · pi(X )

} [1 − exp

{−a[ jq − j(X )]}]

(5.132)

Here, jp is the maximum value of the amount of slippage j and τp is the corresponding shearresistance to j = jp, and jq is the value of j when the shear resistance τ becomes zero in theuntraction process, i.e.

jq = jB −(

τp

k0

)1/n0

(5.133)

178 Terramechanics

Again, the distribution of the shear resistance tms(X ) acting on the four sides of the grousersof the main part of track belt can be calculated from the average normal stress Fx/DH actingon the side part of the track belt given in Eq. (5.14) as follows:

For the untraction state of jq < j(X ) < jp = jB = jfmax

t′ms(X ) = 4H[τp − k0{ jp − j(X )}n0

](5.134)

For the reciprocal traction state of j(X ) � jq

t′′ms(X ) = −4H

{mc + mf

pi(X )

πcot−1

(H

B

)} [1 − exp

{−a[ jq − j(X )]}]

(5.135)

The shear resistance changes from a positive value to a negative one at X = DD where j(X )becomes jq in the untraction state. Here, DD can be calculated as follows:

DD = −( jB − jq)(

1 + 1

i′b

)(5.136)

In the range 0 � X � DD, the amount of slippage j(X ) takes on a positive value, while, inthe range DD < X < D, j(X ) takes on a negative value, so that the drag Tmb and Tms can becalculated as:

Tmb = 2B∫ DD

0τ ′

m(X ) dX + 2B∫ D

DDτ ′′

m(X ) dX (5.137)

Tms =∫ DD

0t′ms (X ) dX +

∫ D

DDt′′ms(X ) dX (5.138)

When jB is less than jfmax, the distribution of shear resistance τ(X ) acting on the main partof the track belt can be calculated for the reciprocal traction state as:

τ ′′′m (X ) = −{mc + mf pi(X )

} ∥∥∥1 − exp[−a{ j′q − j(X )}

]∥∥∥ (5.139)

j′q = jf max − (τ ′p/k0)1/n0

τ ′p = {

mc + mf pi(X )}{

1 − exp(−ajf max)}

Yet again, the distribution of shear resistance tms(X ) acting on the four sides of the grousersof the main part of track belt can be calculated as:

t′′′ms(X ) = −4H

{mc + mf

pi(X )

πcot−1

(H

B

)}=∥∥∥1 − exp

[1a{ j′q − j(X )}

]∥∥∥ (5.140)

In this case, τm(X ), tms(X ) take on negative values for the whole range of the main part oftrack belt. Hence the drag Tmb acting on the base area of grousers of the main part of trackbelt can be calculated as:

Tmb = 2B∫ D

0τ ′′′

m (X ) dX (5.141)

The drag Tms acting on the side part of grousers of the main part of track belt can similarlybe calculated as:

Tms = 2B∫ D

0t′′′m (X ) dx (5.142)


As the above calculations correspond to sf 0 = sf 0i, sr0 = sr0i in the cases of Section 5.1.2 (1),(2), (3), pi(X ) should be calculated for each amount of sinkage distribution s0i(X ) obtainedby substituting P′ = P − Pf into W cos θt0 as mentioned previously.

Next, let us consider the situation where sf 0i < 0 < H < sr0i in Section 5.1.2 (4).In a similar manner, s0i(X ) and pi(X ) can be calculated for W cos θt0 = P and the real

contact length L of the main part of the track belt can be determined by use of Eq. (5.66). Inthe range 0 ≤ X ≤ D−L, pi(X ) and τm(X ) become zero, while, in the range D−L < X ≤ D,j(X ), τ ′

m(X ) and t′ms(X ) may be given as follows:

j(X ) = i′b1 + i′b

(X − D + L)

τ ′′m(X ) = −{mc + mf pi(X )

}[1 − exp{aj(X )}] (5.143)

t′′ms(X ) = −4H

{mc + mf

pi(X )

πcot−1

(H

B

)}[1 − exp{aj(X )}] (5.144)

Thence, the drag Tmb, Tms can be calculated as follows:

Tmb = 2B∫ D

D−Lτ ′′

m(X ) dX (5.145)

Tms =∫ D

D−Lt′′ms(X ) dX (5.146)

Further, let us consider the case where sf 0i > H > 0 > sr0i in Section 5.1.2 (5). Calculatingthe amount of sinkage s0i(X ) and the contact pressure pi(X ) for W cos θt0 = P − Pf in thesame way as before, the real contact length LL of the main part of the track belt can becalculated from Eq. (5.71).

For the condition 0 � X � LL, when jB equals jf max, τ ′m(X ) is given by Eq. (5.131) and

t′ms(X ) is given by Eq. (5.134) when the amount of slippage j(X ) given in Eq. (5.130) is inthe state of untraction jq < j(X ) < jp, while τ ′′

m(X ) is given Eq. (5.132) and t′′ms(X ) is givenfrom Eq. (5.135) when j(X ) is less than jq in the reciprocal traction state. When jB is lessthan jf max, τ ′′′

m (X ) is given by Eq. (5.139) and t′′′ms(X ) is given from Eq. (5.140) when j(X )is in the reciprocal traction state.

On the other hand, for LL < X � D, τ ′′′m (X ) for the reciprocal traction of j(X ) < jq can be

determined by:

τ ′′′′m (X ) = −mc

∥∥1 − exp[−a

{jq − j(X )

}]∥∥ (5.147)

t′′′′ms(X ) = −4Hmc

∥∥1 − exp[−a

{jq − j(X )

}]∥∥ (5.148)

Thence, when jB equals jf max, the value of DD given by Eq. (5.136) is usually in the rangeof 0 � DD � LL, and Tmb and Tms can be calculated as follows:

Tmb = 2B

{∫ DD

0τ ′

m(X ) dX +∫ LL

DDτ ′′

m(X ) dX +∫ D

LLτ ′′′′

m (X ) dX

}(5.149)

Tms =∫ DD

0t′ms(X ) dX +

∫ LL

DDt′′ms(X ) dX +

∫ D

LLt′′′′ms(X ) dX (5.150)

180 Terramechanics

For jB < jf max, the shear resistance τ(X ) becomes negative for the whole range of the mainpart of track belt, so that the drag Tmb and Tms can be determined as:

Tmb = 2B

{∫ LL

0τ ′′′

m (X ) dX +∫ D

LLτ ′′′′

m (X ) dX

}(5.151)

Tms =∫ LL

0t′′′ms(X ) dX +

∫ D

LLt′′′ms(X ) dX (5.152)

(2) Contact part of the front-idlerThe amount of slippage jf (θ) at the contact part of the front-idler during braking action canbe calculated by use of the entry angle θf 0 given in Eq. (5.78) and the angle of inclinationof the vehicle θt0i calculated from Eq. (5.79) by use of the static amount of sinkage offront-idler sf 0i and rear sprocket sr0i as follows:

jf (θ) = (Rf + H )∫ θf 0

θ

{1 − 1

1 + ibcos(θ + θt0i)

}dθ

= (Rr + H )[

(θf 0 − θ) − 1

1 + ib

{sin(θf 0 + θt0i) − sin(θ + θt0i)

}](5.153)

where the amount of slippage at point B, jB can be given as jf (0).The amount of sinkage sf 0i(θ) and the contact pressure pfi(θ) at the contact part of the

front-idler can be calculated from Eq. (5.80) and Eqs. (5.81), (5.82).Then, the shear resistance of the soil τf (θ), tfs(θ) etc. acting on the contact part of the

front-idler can be calculated as,For the traction state of 0 < jf (θ) < jp = jf max

τf (θ) = {mc + mf pfi(θ)} [1 − exp{−ajf (θ)}] (5.154)

tfs(θ) = 4H

{mc + mf

pfi(θ)

πcot−1

(H

B

)} [1 − exp{−ajf (θ)}] (5.155)

For the untraction state of jq < jf (θ) < jp = jf max

τ ′f (θ) = τp − k0{ jp − jf (θ)}n0 (5.156)

tfs(θ) = 4H[τp − k0{jp − jf (θ)}n0

](5.157)

For the reciprocal traction state of jf (θ) < jq

τ ′′f (θ) = −{mc + mf · pfi(θ)} ∥∥1 − exp

[−a{jq − jf (θ)}]∥∥ (5.158)

t′′fs(θ) = −4H

{mc + mf

pfi(θ)

πcot−1

(H

B

)}∥∥1 − exp[−a{jq − jf (θ)}]∥∥ (5.159)

where jp is the maximum value of the amount of slippage j and τp is the corresponding shearresistance to jq. The parameter jq is the value of j when τ becomes zero in the untractionstate.

For the case where cos θf −1 < ib < 0, jf (θ) equals jp at θ = θfp and jf (θ) equals jq atθ = θfq, then the drag Tfb and Tfs acting on the base and side areas of the grousers of the


contact part of the front idler in the direction of the main straight part of the track belt canbe calculated as follows:

Tfb = 2B(Rr + H )[∫ θfq

0

{τ ′′

f (θ) cos θ − pfi(θ) sin θ}

dθ

+∫ θfp

θfq

{τ ′

f (θ) cos θ − pfi(θ) sin θ}

dθ +∫ θf 0

θfp


}dθ

]

(5.160)

Tfs = (Rr + H )

{∫ θfq

0t′′fs cos θ dθ +

∫ θfp

θfq

t′fs(θ) cos θdθ +∫ θf 0

θfp

tfs(θ) cos θ dθ

}(5.161)

Alternately, in the case where ib < cos θf −1, the shear resistance τf (θ) and tfs(θ) can begiven as:

τf (θ) = {mc + mf · pfi(θ)

} [1 − exp{−ajf (θ)}] (5.162)

tfs(θ) = 4H

{mc + mf

pfi(δ)

πcot−1

(H

B

)} [1 − exp{−ajf (θ)}] (5.163)

and therefore,

Tfb = 2B(Rf + H )∫ θf 0

0


}dθ (5.164)

Tfs = (Rf + H )∫ θf 0

0tfs(θ) cos θ dθ (5.165)

The above calculations have been presented for situations corresponding to cases (1), (2),(3) and (5) in Section 5.1.2. For the other case (4), both the values of Tfb and Tfs become zero.

(3) Part of rear sprocketThe amount of slippage jr(δ) of the contact part of the rear sprocket during braking actioncan be calculated by use of the exit angle θr0 = θt0i and the angle of inclination of the vehicleθt0i. The inclination angle is given in Eq. (5.79) from the static amount of sinkage of thefront-idler sf 0i and the rear sprocket sr0i. The overall calculation is then as follows:

jr(δ) = (Rr + H ){

(θt0i − δ) − 1

1 + ib(sin θt0i − sin δ)

}+ i′bD

1 + i′b+ jB (5.166)

The amount of slippage jD at the point D can be given as jr(0) from above equation.The amount of sinkage sr0i(δ) and the contact pressure pri(δ) of the contact part of the

rear sprocket can be calculated through the use of Eqs. (5.88), (5.89) and (5.90).Since we know that the amount of slippage jr(δ) is usually negative, the shear resistance

of the soil τr(δ) and trs(δ) acting on the contact part of the rear sprocket can be calculated as:

τr(δ) = −{mc + mf pri(δ)}

[1 − exp{ajr(δ)}] (5.167)

trs(δ) = −4H

{mc + mf

pfi(δ)

πcot−1

(H

B

)}[1 − exp{−ajr(δ)}] (5.168)

182 Terramechanics

Then, the drag Trb and Trs acting on the base and side areas of the grousers of the contact partof the rear sprocket in the direction of the main part of the track belt BC can be calculatedas follows:


0{τr(δ) cos(θt0i − δ) + pri(δ) sin(θt0i − δ)} dδ (5.169)

Trs = (Rr + H )∫ θr0

0trs(δ) cos(θt0i − δ) dδ (5.170)

The above calculations have been developed for the cases of (1), (2), (3) in Section 5.1.2.For case type (4), Trb and Trs can be calculated by substituting the next amount of slippagejr(δ) and the amount of sinkage sr0i(δ) given in Eq. (5.96) into Eqs. (5.91), (5.92) andEqs. (5.93), (5.94) respectively.

jr(δ) = (Rr + H ){

(θt0i − δ) − 1

1 + ib(sin θt0i − sin δ)

}+ i′bL

1 − i′b(5.171)

On the other hand, for case type (5), Trb and Trs can be calculated by substituting τr(δ)given Eq. (5.167) and trs(δ) given in Eq. (5.103) [which should be modified in considerationof the fact that the term of mc acts on the contact part of rear sprocket even if the contactpressure pri(δ) equals zero] into Eqs. (5.169) and (5.170) respectively.


As mentioned in Section 5.2.4, when a rigid tracked vehicle is running on a horizontal softterrain, the compaction resistance T2 can be calculated in the manner shown in Eq. (5.103)by use of the amount of sinkage sri of the rear sprocket. Several methods for the calculationof sri for various situations will now be presented:

(1) For the case where 0 ≤ sf 0i ≤ sr0i

The total amounts of sinkage sfi, sri can be calculated by substituting (a) the static amountsof sinkage sf 0i and sr0i given in Eq. (5.46) – which are normal to the main part of the rigidtrack belt, (b) the amounts of slip sinkage sfs, srs given in the next equations and (c) theangle of inclination of the vehicle θti given in Eq. (5.52) into Eqs. (5.97) and (5.98).

The amount of slip sinkage sfs at the point B on the front-idler can be calculated bysubstituting the contact pressure pfi(θm) and the amount of slippage of the soil jfs into theprevious Eq. (4.8) as follows;

sfs = c0

M∑m=1

{pfi(θm) cos θm

}c1

{(m

Mjfs

)c2

−(

m − 1

Mjfs

)c2}

(5.172)

where

θm = θf 0

(1 − m

M

)jfs = −(Rf + H )ib sin θf (5.173)


The amount of slip sinkage srs at the point C on the rear sprocket can be calculated bysubstituting the contact pressure pi(X ) and the amount of slippage of the soil js into theprevious Eq. (4.8) as follows:

srs = sfs + c0

N∑n=1

{pi

(nD

N

)}c1[(

njsN

)c2

−{

(n − 1)jsN

}c2]

(5.174)

where

Js = −ibD

(2) For the case where sf 0i > sr0i > 0The total amount of sinkage of the rear sprocket sri following the passage of the front wheelshould be larger than the total amount of sinkage sfi of front-idler. Therefore, sri should becalculated as in the following equation, while sfi is given from Eq. (5.97):

sri = (sf 0i + srs) cos θti (5.175)

(3) For the case where sf 0i < 0 < H < sr0i

The total amounts of sinkage of front-idler sfi and rear sprocket sri can be calculated bysubstituting the amounts of slip sinkage sfs and srs as shown in the following equations intoEqs. (5.97) and (5.98) respectively:

srs = c0

N∑n=1

{pi

(nL

N

)}c1[(

njsN

)c2

−{

(n − 1)jsN

}c2]

(5.176)

where

Js = −ibL

(4) For the case where sf 0i > H > 0 > sr0i

The total amount of sinkage of the front-idler sfi and the rear sprocket sri can be calculatedby substituting the amount of slip sinkage sfs given in Eq. (5.172) and srs as shown in thefollowing equation, into Eqs. (5.97) and (5.175) respectively.

srs = sfs + c0

N∑n=1

{pi

(nLL

N

)}c1[(

njsN

)c2

−{

(n − 1)jsN

}c2]

(5.177)

where

Js = −ibLL

5.3.5 Energy equilibrium analysis

The effective input energy E1 supplied by a braking torque to the rear sprocket of a movingtracked-vehicle of necessity equates to the total output energies i.e. to the sum of thecomponent elements namely: the compaction energy E2 required make a rut under thetrack belt, the slippage energy E3 that occurs due to the shear deformation of the soil atthe interface between the track belt and the terrain and the effective braking force energy

184 Terramechanics

E4 which becomes the braking work done on the vehicle. The resultant energy equilibriumequation can then be expressed as:

E1 = E2 + E3 + E4 (5.178)

When the track belt moves one contact length of the track belt D relative to the vehicleitself, the vehicle is transferred horizontally to the position of D cos θti/(1+ i′b). Under theseconditions the various energy components can be calculated as in the following equations:

E1 = T1D = T3D (5.179)

E2 = T2Dcos θti

1 + i′b= T2D

i

1 + ib(5.180)

E3 = T3Di′b

1 + i′b+ WD

1

1 + i′bsin θti = T3D

(1 + ib) cos θti − 1

(1 + ib) cos θti+ WD

1

1 + ibtan θti

(5.181)

E4 = T4Dcos θti

1 + i′b= T4D

1

1 + ib(5.182)

Also, each energy component per unit time can be expressed in terms of the vehicle speedV and the circumferential speed of the track belt V ′ as follows:

E1 = T1V ′ = T3V ′ = T3V (1 + ib) (5.183)

E2 = T2V ′ 1

1 + ib= T2V (5.184)

E3 = T3V ′ (1 + ib) cos θti − 1

(1 + ib) cos θti+ WV ′ 1

1 + ibtan θti

= −T3V

{1

cos θti− (1 + ib)

}+ WV tan θti (5.185)

E4 = T4V ′ 1

1 + ib= T4V (5.186)

5.3.6 Effective braking force

The effective braking force T4 developed by a rigid tracked vehicle running during brakingaction can be obtained by substituting the drag T3, the compaction resistance T2 and theangle of inclination of the vehicle θti into Eq. (5.128). In the T4−ib curve, the ‘optimum skid’ibopt is defined as the skid when the effective input energy |E1| calculated under a constantvehicle speed V takes a maximum value and the ‘optimum effective braking force’ T4opt isdefined as the corresponding effective braking force with ibopt . The ‘braking efficiency’ Eb

is defined as:

Eb = E4

E1= VT4

V ′T1= 1

1 + ib· T4

T1(5.187)

Figure 5.14 shows a flow chart that may be used to calculate the trafficability of a rigidtracked vehicle running on a soft terrain during braking action. At the outset, the vehicledimensions and the terrain-track system constants are required as input data. After that, the


Figure 5.14. Flow chart to calculate braking performance of rigid tracked vehicle running on softterrain during braking action.

contact pressure distribution pf 0, pr0 and p0(X ) of the rigid tracked vehicle at rest can becalculated for the static amount of sinkage distribution s0(X ) and the angle of inclinationof the vehicle θt0 in each case of (1) ∼ (5) as mentioned in Section 5.1.2, and it should beiteratively calculated until the amount of sinkage s0(X ) is strictly determined. Next, theskid ib during braking action can be computed from ib = −1% to −99% successively. Theflow system of the calculation can be divided into the following three streams on the basisof the static amounts of sinkage of the front-idler sf 0i and the rear sprocket sr0i.

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For the cases of sf 0i ≥ H , sr0i ≥ H ; 0 ≤ sf 0i < H < sr0i; sf 0i > H > sr0i ≥ 0, the parameterss0(X ) and θt0i can be calculated from Eqs. (5.46) and (5.79) respectively, and then the factorspfi, pri, pi(X ) can be calculated from Eqs. (5.47) and (5.48). After that, the amount of slipsinkage sfs and srs can be calculated from Eqs. (5.172) and (5.174) respectively. Thence, thetotal amount of sinkage of the front-idler sfi and the rear sprocket sri can be calculated fromEqs. (5.97) and (5.175) respectively in the case of 0 � sf 0i � sr0i, and from Eqs. (5.97) and(5.175) respectively in the case of sf 0i > sr0i > 0. After that, si(X ) and θti can be determinedby use of Eqs. (5.102) and (5.52) respectively.

As a next step, the compaction resistance T2 can be calculated by substituting sri intoEq. (5.103). The drag T3 can then be calculated from Eq. (5.59) as the sum of Tmb givenin Eq. (5.137) or (5.140), Tms given in Eq. (5.138) or (5.141), Tfb given in Eq. (5.160),Tfs given in Eq. (5.161), Trb given in Eq. (5.169) and Trs given in Eq. (5.170).

Furthermore, the braking force T1 given in Eq. (5.127), the effective braking force T4

given in Eq. (5.128) and the ground reaction P given in Eq. (5.54) should be iterativelycalculated until the drag T3 is uniquely determined.

It should be noted, in the process of calculation of the static amount of sinkages of thefront-idler and the rear sprocket, that the ground reaction P′ acting on the main part ofthe rigid track belt should be calculated by subtracting the ground reaction Pf acting on thecontact part of the front-idler from the above mentioned ground reaction P.

For the case of sf 0i < 0 < H < sr0i, the factors s0i(X ) and θt0i and the parameters pfi

(=0), pri, pi(X ) can be calculated in a similar manner. The amount of slip sinkage of therear sprocket srs can be calculated from Eq. (5.176), while that of the front-idler sfs iszero. Thence, the total amount of sinkage of the front-idler sfi and the rear sprocket sri

can be calculated from Eqs. (5.97) and (5.98) respectively and then si(X ) and θti can bedetermined.

Next, the compaction resistance T2 can be calculated from the previously obtainedamount of sinkage sri.

The drag T3 can be calculated as the sum of Tmb given in Eq. (5.145), Tms given inEq. (5.146), Trb given in Eq. (5.92) and Trs given in Eq. (5.94). The braking force T1, theeffective braking force T4 and the ground reaction P can be iteratively calculated until thedrag T3 is strictly determined.

For the situation where sf 0i > H > 0 > sr0i, the factors s0i(X ) and θt0i and the parameterspfi, pri (=0), pi(X ) can be calculated in the same way. The amount of slip sinkage of thefront-idler sfs can be calculated from Eq. (5.172) and the amount of slip sinkage of the rearsprocket srs can be calculated from Eq. (5.177). Thence, the total amount of sinkage ofthe front-idler sfi and the rear sprocket sri can be calculated from Eqs. (5.97) and (5.175)respectively and then si(X ) and θti can be determined.

Following this, the compaction resistance T2 can be calculated from the previouslyobtained amount of sinkage sri. The drag T3 can be calculated from Eq. (5.59) as the sumof Tmb given in Eq. (5.149) or (5.151), Tms given in Eq. (5.150) or (5.152), Tfb given inEq. (5.160), Tfs given in Eq. (5.161), Trb given in Eq. (5.169) and Trs given Eq. (5.170). Thebraking force T1 given in Eq. (5.127), the effective braking force T4 given in Eq. (5.128)and the ground reaction P given in Eq. (5.54) can be iteratively calculated until the drag T3

is uniquely determined.For this condition, in the process of calculation of the static amount of sinkage of the

front-idler and the rear sprocket, the ground reaction P′ acting on the main part of the rigid


track belt should be determined by subtracting the ground reaction Pf acting on the contactpart of the front-idler from the ground reaction P calculated above.

By use of the output values so obtained i.e. T1, T2, T3, T4, P, sfi, sri and θti, the variousenergy components given in Eqs. (5.179) ∼ (5.186) as well as the braking efficiency Eb

given in Eq. (5.187) can be determined. Also, the distributions of the amount of slippagej(X ), contact pressure pi(X ), shear resistance τi(X ) under the track belt and the distributionsof the amount of sinkage of the track belt si(X ) can be calculated.

The above calculation can be sequentially carried out for various skids ib, so thatT1 ∼ T4 − ib curves, sfi, sri − ib curves, ei, θti − ib curves, E1 ∼ E4 − ib curves and Eb − ibcurve can be plotted. At the same time, the optimum effective braking force T4opt , theoptimum skid ibopt can be determined.

5.4 EXPERIMENTAL VALIDATION

The validity of this proposed simulation method for analysis of the trafficability of a rigidtracked vehicle has been confirmed against an experimental test study [9] that used a modelrigid tracked vehicle running on a remolded silty loam terrain. The water content of thesilty loam was approximately 30% and the cone index of the terrain was approximately31 kPa. The track model was fitted with standard T shaped steel grousers of track width20 cm and grouser height H = 3.2 cm as shown in Figure 5.15.

The terrain-track system constants obtained from the plate loading test and the platetraction test for the track model plate on the silty loam terrain have previously been givenin Table 4.1.

Figure 5.16 shows a plan and side-elevation view of the model rigid-tracked vehicle. Thevehicle had a rear wheel drive system comprised of a constant speed motor of 1.5 kW.

The track belt was made of roller chain attached with the same track plate as the modeltrack plate.

As shown in Photo 5.1, this rigid tracked vehicle used a multi-roller system instead of theroad roller system which is in general use. The weight of the model rigid tracked vehiclewas 3.55 kN. The geometrical details of the vehicle and other specifications are given inTable 5.1.

Figure 5.17 is a schematic of the experimental apparatus. It shows the positions of thetest vehicle on the surface of the soil bin during driving and braking action. The size of thesteel panel soil-bin was of length 540 cm with a width of 150 cm and a height of 60 cm.

Figure 5.15. Dimensions of track model plate (standard T shaped grouser).

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Figure 5.16. Outline of test vehicle.

Photo 5.1. Model rigid tracked vehicle.


Table 5.1. Specification and dimension of rigid tracked vehicle.

Vehicle weight W (kN) 3.55Average contact pressure pm (kPa) 12.5Height of center of gravity G from bottom of track belt hg (cm) 35.3Output of motor (kW) 1.5Contact length of track belt D (cm) 71Circumferential length of track belt (cm) 236Clearance A (cm) 13.5Radius of frontidler Rf (cm) 14.8Radius of rear sprocket Rr (cm) 14.8Track width B (cm) 20Circumferential speed of track belt during driving action V ′ (cm) 9.4Eccentricity of center of gravity of vehicle e (%) −0.5Vehicle speed during braking action V (cm/s) 9.4Grouser pitch Gp (cm) 5.1 10.2 20.4Grouser height H (cm) 3.2Gp/H 1.6 3.2 6.4Distance between application point M of effective tractive ld (cm)effort and central line of vehicle 50.8

Height of application point M of effective tractive effort hd (cm) 32.5Distance between application point N of effective braking lb (cm)

force and central line of vehicle −17.3Height of application point N of effective braking force hb (cm) 11.5Central interval of track belt C (cm) 67.2Thickness of track belt (cm) 0.3

Figure 5.17. Experimental apparatus and vehicles during driving and braking action.

The silty loam test soil was deposited uniformly until a depth of 40 cm was attained. Thesoil properties were as follows: real specific gravity 2.84, liquid limit 33.2%, plastic limit21.4%, plastic index of 11.8%, average grain size of 54 µm, coefficient of curvature of0.31, coefficient of uniformity 6.40, bulk density of 18.6 kN/m3, void ratio of 0.85, coneindex of 31 kPa and water content of 29.5 ± 1.0%. The silty loam sample was remolded tohave a uniform strength in the vertical direction of the soil bin. After flattening the surfaceof the test soil with a grader, several trafficability tests were executed.

For this series of tests, the soil sample and the soil bin were the same ones as were usedfor the plate loading and traction test of the previous track model plate. A tractive apparatus

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was made to control the vehicle speed and slip ratio or skid during driving or braking actionrespectively. The power source of the tractive system was a variable speed motor of 3.7 kW.The wire speed out of the winch of the tractive device could be controlled to a range ofspeeds V from 3.9 cm/s to 23.6 cm/s. The effective tractive effort or the effective brakingforce was measured using a load cell of maximum capacity 20 kN. The load cell had asensitivity of 1 N and was connected to the wire and to the vehicle as shown in the diagram.

The experimental method that was employed was as follows. During driving action, thefeed speed of the wire, i.e. the vehicle speed V , was controlled to a value somewhat lessthan the circumferential speed V ′ of the track belt depending on the various slip ratios idemployed. For braking action, the vehicle speed V is required be controlled to be largerthan the track speed V ′ depending on the various skids ib as shown in the diagram. For eachtest, the driving force or braking force T1, the effective driving or braking force T4, theinitial amounts of sinkage of the front-idler sf 0 and the rear sprocket sr0 and the steady stateamounts of sinkage of the front-idler sfi and the rear sprocket sri are measured. It is notedthat the factor T1 can be calculated from the measured driving or braking torque divided bythe radius of the rear sprocket after subtracting the internal friction between the track beltand the multi-rollers. T4 can be measured by use of the load cell as mentioned previously.In the test set up this value was recorded automatically.

For a rigid tracked vehicle, Figure 5.18 shows the experimentally derived relationshipsbetween T1, T4 and slip ratio i for a machine having several track belts of alternate grouser

Figure 5.18. Relationship between driving and braking force T1, effective driving and braking forceT4 and slip ratio or skid i.


pitches Gp = 5.1, 10.2 and 20.4 cm respectively. For both the driving and braking state,the larger the slip or skid, the larger are the magnitudes of |T1|, |T4| that are developed.At the same time, the compaction resistance T2 increases also with increasing values of|i|. It is observed that the factors, |T1|, |T4| had a maximum value with a grouser pitchGp = 10.2 cm.

The developed driving force T1 is less than the braking force |T1| for the same slip ratioand skid |i|. This results from the fact that the amount of slippage at the rear end of the trackbelt during driving action is smaller than that during braking action. As a consequence, theshear resistance is not sufficiently mobilized in the driving state. In quantitative terms thiseffect can be seen in that the amount of slippage j at the rear end of the track belt of contactlength D during braking action is equal to ibD/(1 + ib) from Eq. (5.124), while the amountof slippage j during driving action is equal to idD from Eq. (5.43).

Figure 5.19 shows the experimentally derived set of relations that prevail between theamount of sinkage of the front-idler sfi, the amount of sinkage of the rear sprocket sri andthe slip ratio i for various values of Gp. During driving action, the amount of sinkage ofthe rear sprocket sri increases with the increasing values of slip ratio while the amount ofsinkage of the front-idler sfi takes on negative values i.e. the position of the front-idler tendsto raise up with slip ratio so that the angle of inclination of the vehicle θti increases with i.

During braking action, both the amounts of sinkage sfi and sri increase with the increasingvalues of skid |ib|. The amount of sinkage of the rear sprocket sri is always larger than thatof front-idler sfi due to the increasing amount of slippage.

Figure 5.19. Relationship between amounts of sinkage of front idler sfi, rear sprocket sri and slipratio or skid i.

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Photo 5.2. Rut of track belt during driving action (id = 49%, Gp = 10.2 cm).

Photo 5.3. Rut of track belt during braking action (ib = −50%, Gp = 10.2 cm).

The amount of sinkage of the rear sprocket i.e. the rut depth of the vehicle sri during drivingaction takes on larger values than those during braking action for the same slip ratio andskid |i|. This is because the amount of slippage js of soil in the passage of the vehicle duringdriving action equates to idD/(1− id) as shown in Eq. (5.45) rather than that during brakingaction of ibD as shown in Eq. (5.126).

These phenomena can be verified from a detailed study of the shape of the rut depth asshown in Photo 5.2.

This photo was taken during driving action at a slip ratio of id = 49%. Photo 5.3 wastaken during braking action at a skid of ib = −50%.

Moreover, the rut depth sri during driving action took its maximum value at Gp = 5.1 cm.The nearest follower was Gp = 20.4 with 10.2 cm next. The rut depth sri during brakingaction reached a maximum value when Gp = 10.2 cm followed by Gp = 20.4 and 5.1 cmrespectively.

Figure 5.20 shows the experimental relations that developed between the angle of incli-nation of the vehicle θti and the slip ratio during driving action or with skid during brakingaction i. The tests were for a machine operating in a running steady-state. The values for


Figure 5.20. Relationship between angle of inclination of vehicle θti and slip ratio or skid i.

Photo 5.4. Model rigid tracked vehicle running during driving action (id = 49%, Gp = 5.1 cm).

the angle θti are larger during driving action than during braking action, so that the positionof the front-idler rises up beyond the surface of terrain and consequently the eccentricityof ground reaction exceeds 1/6.

This process is confirmed in Photo 5.4. The photo was taken during driving action. Asan example of a simulation analysis of the trafficability of a rigid tracked vehicle [10],

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Figure 5.21. Relationship between driving and braking force T1, effective driving and braking forceT4 and slip ratio or skid.

the tractive and the braking performance of a model rigid tracked vehicle having the spec-ification as shown in Table 5.1 on the terrain having the terrain-track system constants asshown in the previousTable 4.1 was simulated for both driving and braking action. The vehi-cle dimensions and the terrain-track system constants for the grouser pitch of Gp = 10.2 cmwere used as input data in the flow chart as shown in Figures 5.9 and 5.14. Here, there wasno need to consider the size effect of the track belt on the terrain-track system constantsbecause the contact area of the model track belt as shown in Figure 5.15 was almost thesame value as that of the model rigid tracked vehicle.

Figure 5.21 shows the comparison between the measured and theoretical value in therelationship between the driving and braking force T1, the effective driving and brakingforce T4 and the slip ratio or skid i. Both the measured and theoretical results agree very wellwith each other. Also, the analytical results are well verified by the experimental data. Forall values of slip ratio or skid i, the difference between the driving force T1 and the effectivedriving force T4 became larger than the difference between the braking force |T1| and theeffective braking force |T4|. This is because the compaction resistance T2 calculated fromthe amount of slip sinkage due to the amount of slippage during driving action becomeslarger than that during braking action, and the component of the ground reaction P inthe direction of T4, e.g. P sin θti during driving action increases with the correspondingincrement of the angle of inclination of vehicle as shown in Figure 5.20 rather than thatduring braking action. That is, the sum of T2 and Psin θti in Eq. (5.49) during driving actionbecomes larger than that during braking action.

Figure 5.22 shows the comparison between the measured and theoretical value in therelationship between the amount of sinkage of front-idler sfi, rear sprocket sri and slip ratioduring driving action or skid during braking action i. In these cases, the analytical resultsagree well with the experimental results. The amount of sinkage of rear sprocket sri duringdriving action became almost twice in comparison with that during braking action. This isbecause the amount of slippage js of soil in the passage of the vehicle during driving action


Figure 5.22. Relationship between amounts of sinkage of front idler sfi, rear sprocket sri and slipratio or skid i.

Figure 5.23. Relationship between eccentricity of ground reaction ei and slip ratio or skid i.

becomes larger than that during braking action, as mentioned previously. Moreover, it canbe explained theoretically that the amount of slip sinkage increases remarkably when theslip ratio during driving action approaches 100%.

Figure 5.23 shows the analytical relationship between the eccentricity ei of the groundreaction and slip ratio during driving action or skid during braking action i respectively.The eccentricity during driving action increases with the increment of slip ratio and thereis some ranges showing ei > 1/6, while the eccentricity during braking actions becomesless than 0.05. These phenomena correspond well with the tendencies of the amounts ofsinkage of the front-idler and rear sprocket.

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Figure 5.24. Relations between energy components E and slip ratio or skid i during passage of trackbelt D.

Figure 5.25. Relationship between energy components E and slip ratio i during driving action(extended diagram).

Figure 5.24 shows the analytical relationships pertaining to each of the energy componentsE1, E2, E3 and E4 during the passage of the track belt of the main contact length D aroundthe vehicle and slip ratio i. Comparing each energy component during driving action, E2,E3 and E4 during braking action increases remarkably with increasing values of skid |i|.This is because E2, E3 and E4 take on an infinite value in Eqs. (5.180) and (5.182) whenthe skid ib approaches −100%, while E1 approaches a constant value in Eq. (5.179). Inthe calculation of E2, E3 and E4 during braking action, the moving distance of the vehiclebecomes D/(1 + ib) which is also very large comparing with the moving distance of vehicleD(1 − id) during driving action. Figure 5.25 is an enlarged diagram of E1 ∼ E4 − i relationsduring driving action of Figure 5.24. E3 increases parabolically with the increases in theslip ratio id , while E1 approaches a constant value. On the other hand, E2 and E4 takesa maximum value at some slip ratio respectively. A tracked vehicle can develop a maximum


Table 5.2. Dimensions of rigid tracked vehicle.

Vehicle weight W (cm) 40.0Average contact pressure pm (kPa) 23.0Height of center of gravity G from bottom of track belt hg (cm) 50Contact length of track belt D (cm) 170Width of track belt B (cm) 50Radius of frontidler Rf (cm) 25Radius of rear sprocket Rr (cm) 25Vehicle speed during braking action V (cm/s) 100Grouser height H (cm) 6.5Grouser pitch Gp (cm) 14.6Eccentricity of center of gravity e 0.00Distance between application point of effective tractive

effort and braking force and central line of vehicle l (cm) 120Height of application point of effective tractive effort and braking force h (cm) 30Circumferential speed of track belt during driving action V ′ (cm/s) 100

tractive work at an optimum slip ratio idopt when E4 takes a maximum value. In this case,the effective driving force T4opt is 2.44 kN at the optimum slip ratio of iopt = 28%, and themaximum effective driving energy E4 is 166.6 kNcm.

The usefulness of the application of the simulation analytical method as presented in theprevious Section can be verified from the experimental results of the model rigid trackedvehicle. Using this method, it is possible to understand properly the tractive and brakingperformance of a given bulldozer running on any kind of soft terrain. In future, it will beshown that this simulation analytical method is valid and useful for design and productionof new bulldozers for running on soft terrain.

5.5 ANALYTICAL EXAMPLE

5.5.1 Pavement road

In the following section, we present some simulation analysis results for a rigid trackedvehicle of weight 40 kN running during driving and braking action on a flat concrete pave-ment. The simulation has been carried out by use of the flow chart given in Figures 5.9and 5.14. The general properties of the vehicle are as shown in Table 5.2. The terrain-tracksystem constants for the equilateral trapezoidal rubber grousers of base length L = 3 cmand for the concrete pavement system are as given in Figure 4.6 and the previous Table 4.3.

For the simulation, it is assumed that the pavement road behaves as an elastic material, thatthe terrain-track system constant k1 equals 9.8 × 104 N/cmn1+1, and that n1 = 1, k2 = n2 = 0and c0 = c1 = c2 = 0. The size effect of the constants fs, fm, jm is not considered.

In terms of the outcomes of running the simulation are concerned, let us now examinethe results for the driving state. We will look at the braking state results a little later.

Figure 5.26 is a plot of the relations between the driving force T1, the effective drivingforce T4 and the slip ratio id . For this case, T4 works out to be equal to T1 since thecompaction resistance T2 becomes zero for the negligibly small amount of sinkage of thetrack belt. T4 shows an almost constant value, after it takes an early maximum value of41.25 kN at an optimum slip ratio of iopt = 1%.

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Figure 5.26. Relationship between driving force T1, effective driving force T4 and slip ratio id duringdriving action (concrete pavement road).

Figure 5.27. Relationship between eccentricity of ground reaction and slip ratio during driving action(concrete pavement road).

Figure 5.27 shows the relationship between the eccentricity ei of the ground reaction andthe slip ratio id . The eccentricity ei takes up an almost constant value, after it passes a peakvalue of 0.0303 at idopt = 1%. Figure 5.28 shows the relationships between the individualenergy components E1, E2, E3, E4 and slip ratio id . E1 stays at an essentially constant valuedue to the constant circumferential speed of the track belt V ′. The component E2 doesnot occur so no land locomotion resistance develops. The component E3 increases almostlinearly with increasing values of slip ratio id . E4 decreases almost linearly with id to a valueof zero at id = 100% after it takes a peak value of 4083 kNcm/s at idopt = 1%.

Additionally, as shown in Figure 5.29, the tractive efficiency Ed becomes very highand peaks at a maximum value of 98.9% at idopt = 1%. After this maximum it decreasesgradually and almost linearly with id to a value of zero at id = 100%.


Figure 5.28. Relationship between energy components E1 to E4 and slip ratio during driving action(concrete pavement road).

Figure 5.29. Relationship between tractive efficiency E4 and slip ratio i4 during driving action(concrete pavement road).

Figure 5.30 shows the longitudinal distribution of normal pressure for id = 10, 20 and 30%and the longitudinal distributions of shear resistance for id = 1, 5, 10, 20 and 30%. Thenormal pressure distribution increases monotonically toward the rear end of the track belti.e. the rear sprocket and it increases irrespective of the slip ratio. On the other hand, theshear resistance distribution varies quite dramatically with the amount of slippage for smallmagnitudes of slip ratio id . It tends, though, to show a constant Hump type curve having apeak value just under the front-idler for large magnitudes of slip ratio id .

Following these considerations of the driving state, the simulation results for the brakingstate are now presented.

Figure 5.31 shows the relations between the braking force T1, the effective braking forceT4 and the skid ib. In this case, T4 also computes to be equal to T1 since the compactionresistance T2 becomes zero for the negligibly small amount of sinkage of the track belt thatoccur in this case. The braking force T4 takes on an almost constant value, after it reachesan early minimum value of −41.78 kN at an optimum skid ibopt = −1%.

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Figure 5.30. Distribution of contact pressure during driving state (concrete pavement road).

Figure 5.31. Relationship between braking force T1 and effective braking force T4 and skid ib duringbraking action (concrete pavement road).

Figure 5.32 shows the relationship between the eccentricity ei and the skid ib. Theeccentricity ei is essentially constant value, after it takes a peak value of −0.0307 atibopt = −1%.

Figure 5.33 shows the relationship between the energy components E1, E2, E3, E4 andskid ib. |E1| decreases almost linearly with increasing values of |ib| to a value of zeroat ib = −100%. Prior to this however it has peaked at a maximum value of 4139 kNcm/sat ibopt = −1%. E2 does not occur in this case and as a consequence no land locomotionresistance occurs. The component E3 increases almost linearly with the increasing values ofskid |ib|. E4 shows a constant value due to the constant vehicle speed V . As a consequenceof these conditions, the braking efficiency Eb becomes very high as shown in Eq. (5.187)and increases hyperbolically with increasing values of |ib|.

Figure 5.34 shows the longitudinal distributions of normal stress at ib = −10, −20% andthe longitudinal of shear resistance at ib = −1, −5, −10 and −20%. The normal pressuredistribution decreases monotonically toward the rear end of the track belt e.g. the rearsprocket and it decreases irrespective of the value of the skid.


Figure 5.32. Relationship between eccentricity ei of ground reaction and skid ib during braking action(concrete pavement road).

Figure 5.33. Relationship between energy components E1 ∼ E4 and skid ib during braking action(concrete pavement road).

Figure 5.34. Distribution of contact pressure during braking state (concrete pavement road).

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Figure 5.35. Relationship between effective tractive effort (D.B.P.) of over-snow vehicle and slip ratioid running on various snow covered terrains [14].

On the other hand, the shear resistance distribution varies quite dramatically with amount ofslippage for small magnitudes of skid |ib|. However, it tends to show a constant Hump typecurve having a peak value just under the front-idler for the large magnitudes of skid |ib|.


In this section, the results of an energy analysis [11] for a tracked over-snow vehicle runningon a snow covered terrain composed of a newly fallen dry snow and sintered snow underlow temperature of −13 ◦C will be presented. The findings are based on a study [12] ofrectangular plate loading tests and a series [13] of vane cone tests carried out on a snowcovered terrain.

When the engine power of a tracked over-snow vehicle is large enough to develop adriving torque at the rear sprocket greater than the maximum thrust occurring at the interfacebetween the track belt and the snow covered ground, the effective driving force of the trackedover-snow vehicle takes a maximum value at a slip ratio of zero and after that it decreasesrapidly with increasing slip ratio, as shown in Figure 5.35 [14]. Here, the compression andshear deformation characteristics of each snow sample of A0 ∼ D14 exhibit a rigid plasticbehaviour under low temperature, as presented previously in Sections 1.3.2 and 1.3.3. Inthis case, the maximum contact pressure of the rigid track belt of contact length 285 cmand track width of 74 cm is calculated to be 33.9 kPa.

Additionally, the effective driving force of a tracked over-snow vehicle running on variouskinds of snow covered terrain composed of different snow materials decreases linearly withincreases in the compressive deformation energy of snow for any depths of deposited snow.That is, as shown in Figure 5.36, the effective driving force D.B.P. (kN) can be expressed[15] as a function of the maximum driving force T (kN) transmitted from the engine, thecontact length of the track belt D (m) and the compressive deformation energy ED (kNm)of the snow. That is:

D.B.P. ≈ T − ED/D (5.188)


Figure 5.36. Relationship between effective tractive effort D.B.P. of over-snow vehicle and compres-sive deformation energy ED for various snow covered terrains (T : Maximum driving force).

where ED is the penetration work of the rigid track belt into the snow until the contactpressure reaches a maximum for the track belt.

In general, ED increases with increasing depth of snow and the effective tractive efforti.e. the effective driving force of the tracked over-snow vehicle decreases with increases insnow-thickness.

Next, let us present some results for an energy analysis [16] of a tracked over-snow vehiclerunning on a snow covered terrain of a depth of 10 cm which is composed of wet snow at thetemperature of 0 ◦C. The analysis is based on the results of circular plate loading tests andring shear test results that have been carried out on the snowy terrain. In this case, the traffi-cability of a tracked over-snow vehicle of the contact length 350 cm, the width 50 cm, withan eccentricity in center of gravity of the vehicle of zero and an average contact pressure of8 kPa has been calculated for machine operations under driving action. The results, as givenin Figure 5.37 show the relationships that exist between the various energy components E1,E2, E3, E4 and the slip ratio id . The components E1 and E4 increase parabolically withincreasing values of id . E2 shows an almost constant value. E3 increases slightly with id .

Figure 5.38 displays the calculated relationship between the effective tractive effort D.B.P.and the slip ratio id for several average contact pressure pav. This data is for a tracked over-snow vehicle having the same vehicle dimensions as before. As the average contact pressureincreases, the deposited snow is compressed so that the strength of the snow becomes high.Thence, the maximum effective tractive effort increases and the corresponding slip ratiodecreases at the same time. From this diagram, it can be suggested that there is an optimumaverage contact pressure that maximizes the maximum effective tractive effort for eachaverage contact pressure.

Figure 5.39 indicates that there is an optimum average contact pressure that maximizesthe maximum effective tractive effort for each value of eccentricity in the center of gravityof a tracked over-snow vehicle. This phenomenon has also been observed in connectionwith studies of super weak clayey soil [17]. That is, the maximum draw-bar pull Max. D.B.P.increases with increase in the average contact pressure, but decreases gradually with theincrease in the land locomotion resistance that accompanies the increasing compressivedeformation energy of the deposited snow that develops after the average contact pressure

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Figure 5.37. Relationship between energy components E1 ∼ E4 and slip ratio id (wet snow, depth ofdeposited snow = 10 cm).

Figure 5.38. Relationship between effective tractive effort D.B.P. of over-snow vehicle and slip ratioid for various average contact pressures pav (wet snow, depth of deposited snow 10 cm).

exceeds the optimum average contact pressure. In this case, the optimum average contactpressure occurs somewhere between 27.4 and 42.1 kPa and the maximum value of themaximum effective tractive effort i.e. the maximum draw-bar pull Max. D.B.P. decreaseswith increase in the eccentricity e of the center of gravity of the vehicle.

5.6 SUMMARY

In this chapter, we have set out to analyse the behaviour of the simplest of the trackedvehicle systems – namely that of the rigid-track types – using the model-track method.


Figure 5.39. Relationship between maximum effective tractive effort Max. D.B.P. and average contactpressure for various eccentricities of center of gravity of vehicle (wet snow, depth of deposited snow10 cm).

By use of computer simulation and through the use of shaped-plate traction test data (asdiscussed in Chapter 4) the behaviour of tracked vehicles operating under a number ofdifferent physical circumstances can be predicted and thence analysed.

In this chapter a mix of analytical and empirical methods and parameters have been usedto predict:– The ground pressures under an immobile tracked vehicle as a function of different

inclination angles and extents of track sinkage.– The amount of sinkage, rut depth and drawbar pull that a specific design of tracked

vehicle will produce while it is driving forward or under braking.– The over-ground speed of the vehicle.By systematically varying a machine’s design parameters and/or the ground conditions,trafficability and mobility studies of specific mechanical engineering configurations canbe explored.

REFERENCES

1. The Japanese Geotechnical Society (1982). Handbook of Soil Engineering. pp. 303–343. (InJapanese).

206 Terramechanics

2. Terzaghi, K. (1943). Theoretical Soil Mechanics. pp. 118–143. John Wiley & Sons.3. Akai, K. (1975). Soil Mechanics. pp. 204–209. Asakura Press. (In Japanese).4. Muro, T. (1991). Optimum Track Belt Tension and Height of Application Forces of a Bulldozer

Running on Weak Terrain. J. of Terramechanics, Vol. 28, No. 2/3, pp. 243–268.5. Hata, S. (1987). Construction Machinery. pp. 76–91. Kajima Press.6. Muro, T. (1989). Stress and Slippage Distributions under Track Belt Running on a Weak Terrain.

Soils and Foundations, Vol. 29, No. 3, pp. 115–126.7. Muro, T. (1989). Tractive Performance of a Bulldozer Running on Weak Ground. J. of

Terramechanics, Vol. 26, No. 3/4, pp. 249–273.8. Muro, T. (1990). Control System of the Optimum Height of Application Forces for a Bulldozer

Running on Weak Terrain. Proc. of the 1st Symposium on Construction Robotics in Japan,pp.197–206. JSCE et al. (In Japanese).

9. Muro, T., Omoto, K. & Futamura, M. (1988). Traffic Performance of a Bulldozer Running ona Weak Terrain-Vehicle Model Test. J. of JSCE No. 397/VI-9, pp. 151–157. (In Japanese).

10. Muro, T., Omoto, K. & Nagira, A. (1989). Traffic Performance of a Bulldozer Running ona Weak Terrain – Energy Analysis. J. of JSCE, No. 403/VI-10, pp. 1103–110. (In Japanese).

11. Muro, T. &Yong, R.N. (1980). On Trafficability of Tracked Oversnow Vehicle – Energy Analysisfor Track Motion on Snow Covered Terrain. Journal of Japanese Society of Snow and Ice, Vol.42, No. 2, pp. 93–100. (In Japanese).

12. Muro, T. & Yong, R.N. (1980). Rectangular Plate Loading Test on Snow Mobility of TrackedOversnow – Journal of Japanese Society of Snow and Ice,Vol. 42, No. 1, pp. 25–32. (In Japanese).

13. Muro, T. & Yong, R.N. (1980). On Drawbar Pull of Tracked Oversnow Vehicle. Journal ofJapanese Society of Snow and Ice, Vol. 42, No. 2, pp. 101–109. (In Japanese ).

14. Yong, R.N. & Muro, T. (1981) Prediction of Drawbar-Pull of Tracked Over-Snow Vehicle. Proc.of 7th Int. Conf. of ISTVS, Calgary, Canada. Vol. 3, pp. 1119–1149.

15. Muro, T. (1981). Shallow Snow Performance of Tracked Vehicle. Soils and Foundations, Vol. 24,No. 1, pp. 63–76.

16. Muro, T. & Enoki, M. (1983). Trafficability of Tracked Vehicle on Super Weak Ground. Memoirsof the Faculty of Engineering, Ehime University, Vol. X, No. 2, pp. 329–338. (In Japanese).

EXERCISES

(1) Suppose that a bulldozer is running during driving action on a sandy terrain at a slipratio of id = 30%. The contact length of the bulldozer D is 3.5 m. The radius of the rearsprocket Rr is 50 cm and the angular velocity of rotation ωr is π rad/s. Calculate theslip velocity Vs between the track and terrain, and the amount of slippage j(D) at therear end of the bulldozer.

(2) The bulldozer given in previous problem (1) has passed an arbitrary point X in therunning lane. Calculate the transit time td and the amount of slippage js at the point X .

(3) Imagine a bulldozer of weight W = 100 (kN) running during driving action on a softterrain at a slip ratio of id = 20%. The contact length D of the bulldozer is 4 (m) and thewidth of the track B is 45 (cm). It is observed that the amount of sinkage at the front-idler sf was 3 (cm) and that at the rear sprocket sr was 5 (cm). Calculate the thrust Tmb

acting on the base area of the grousers of the main part of track belt, assuming that thedistribution of the ground reaction is trapezoidal and the amount of eccentricity of theground reaction ei is 0.05. Assume that the terrain-track system constant mc = 50 (kPa),mf = 0.458 and a = 0.244 (1/cm) and that the amount of slippage jB at the position ofthe front-idler can be neglected.

(4) Calculate the compaction resistance T2 and the effective tractive effort T4 of the bull-dozer given in previous problem (3). Assume the height of grouser H = 2.0 cm, the


coefficient of sinkage k1 = 0.763 N/cmn1+1, k2 = 1.491 N/cmn2+1, and the indices ofsinkage n1 = 0.866, n2 = 0.842.

(5) Suppose that the bulldozer given in previous problem (3) is running at a speedV = 100 cm/s. Confirm the fact that at a slip ratio id = 20%, the effective input energyE1 equals the sum of the compaction energy E2, the slippage energy E3 and the effectivetractive effort energy E4.

(6) A tractor of contact length D = 2.5 m is running during braking action on a sandyterrain at a skid value ib = −20%. The circumferential speed of the rear sprocket Rrωr

is 50 πcm/s. Calculate the slip velocity Vs between the tractor and terrain, and theamount of slippage j(D) at the rear end of the tractor.

(7) The tractor given in previous problem (6) has passed an arbitrary point X in the runninglane. Calculate the transit time tb and the amount of slippage js at the point X .

(8) Imagine that a bulldozer of weight W = 100 (kN) is running during braking action ona soft terrain at a skid of ib = −20%. The contact length D of the bulldozer is 400 (cm)and the width of track B is 45 (cm). It is observed that the amount of sinkage of thefront-idler sf is 3.5 (cm) and that of rear sprocket sr is 4.5 (cm). Calculate the drag Tmb

acting on the base area of the grousers of the main part of the track belt, assuming thatthe distribution of the ground reaction is trapezoidal and the amount of eccentricityei of the ground reaction is 0.02. Assume that the terrain-track system constants aremc = 50 kPa, mf = 0.655 and a = 0.258 (1/cm) and that the amount of slippage jB atthe position of front-idler can be neglected.

(9) Calculate the effective braking force T4 and the compaction resistance T2 of thebulldozer given in problem (8). Assume the height of grouser H = 2.0 cm, the coeffi-cient of sinkage k1 = 0.763 N/cmn1+1, k2 = 1.491 N/cmn1+2 and the indices of sinkagen1 = 0.866, n2 = 0.842.

(10) The bulldozer given in previous problem (8) is running at the speed of V = 100 cm/s.Confirm the fact that, at the skid ib = −20%, the effective input energy E1 is equal tothe sum of the compaction energy E2, the slippage energy E3 and the effective brakingforce energy E4.

Chapter 6

Land Locomotion Mechanics of Flexible-Track Vehicles

The common ‘flexible track belt’ system that is mounted on many bulldozers or tractorscomes in two general mechanical engineering styles. In the first style, only up-down move-ments of a track plate that spans between mutually connected road rollers can occur. Thisarrangement is as shown in Figure 6.1. In the second structural configuration or style,up-down and side-side movements of road rollers that are mutually connected by hinges,springs or torsion bars can occur. This second style is shown in Figure 6.2. In general, theflexible track system is appropriate for construction work that must take place on or overterrains that are geometrically rough or which are full of ups and downs.

In the sections that follow, the land locomotion mechanics and some methods of analysisof a flexible tracked vehicle in which the axles of the road rollers are mutually connected,will be presented. Consideration will also be given to the behaviour of the flexible trackbelt under conditions of decreasing initial track belt tension. The analytical methods sodeveloped may also be extendable to the case where vertical and lateral movement of theroad rollers can occur – such as happens in systems which have various styles of suspensionsystem.

6.1 FORCE SYSTEM AND ENERGY EQUILIBRIUM ANALYSIS

Figure 6.3 shows the composite of forces that act on a flexibly tracked vehicle when thevehicle is climbing a slope under driving action or is descending a slope under braking

Figure 6.1. Flexible track belt under mutually-connected road rollers.

Figure 6.2. Flexible track belt under road rollers connected by hinge, spring or torsion bar.

210 Terramechanics

Figure 6.3. Vehicle dimension and several forces acting on several forces during driving (←) orbraking (⇐) action on soft terrain of slope ange β.

action. The slope in this case is defined to have a slope angle β. The symbols and nomen-clature in the diagram are as have been previously introduced in Sections 5.2.2 and 5.3.2.A positive driving torque or a negative braking torque Q acts on the rear sprocket and acorresponding positive driving force or negative braking force T1 acts on the lower contactpart of the track belt or on the upper suspended part of the track belt. The compactionresistance T2 is assumed to act on the front contact part of the track belt at a depth zi and tooperate in a direction parallel to the sloped terrain surface during both driving and brakingactions. The depth zi can be calculated by carrying out a moment balance around the axleof the rear sprocket when an effective braking force i.e. a pure rolling resistance T4 = −T2

is applied to the point F under conditions where the angle of inclination of the vehicle isθ′

ti, the amount of sinkage of the rear sprocket is s′ri and where T1 = T3 = W sin(θ′

ti + β)and P = W cos(θ′

ti + β) for the pure rolling state. The depth can then be calculated as:

zi = s′ri −

[T2

{Rr + (hd − Rr) cos θ′

ti −(

ld − D

2

)sin θ′

ti

}

− W

{hg sin (θ′

ti + β) − D

(1

2− e

)cos(θ′

ti + β)}]/

{T2 + W cos (θ′

ti + β)/ tan θ′ti

}(6.1)

In this equation θ′ti can be calculated from the amounts of sinkage of the front-idler s′

fi andthe rear sprocket s′

ri via the following equation.

θ′ti = sin−1

(s′

ri − s′fi

D

)(6.2)

Flexible-Track Vehicles 211

The positive thrust or negative drag T3, which can be calculated as the sum of the shearresistances that develop at the interface between the soil and track belt, is assumed to actat the tips of grousers and operate in a direction parallel to the main contact part of theflexible track belt. The positive effective driving force or negative effective braking force T4

is assumed to act at a point F in a direction parallel to the sloped terrain surface. Generally,the eccentricity ei of the ground reaction P can be calculated from a moment balance aroundthe axle of the rear sprocket during driving or braking action as follows:

ei = 1

2+ 1

PD

[−T2(Rr − s′

ri + zi) + T4

{(hd − Rr) cos θ′

ti −(

ld − D

2

)sin θ′

ti

}

+ W

{(hg − Rr) sin(θ′

ti + β) − D

(1

2− e

)cos(θ′

ti + β)}]

(6.3)

When a flexibly tracked vehicle is at rest id = ib = 0, T1 = T3 = W sin(θ′ti + β), T2 = T4,

Q = WR sin(θ′ti +β) and the eccentricity e0 of the ground reaction P can be given as follows:

e0 = e + hg − Rr

Dtan(θ′

ti + β) (6.4)

Considering force equilibrium conditions between T1, T2, T3, T4, P and W in the paralleland normal directions to the surface of the sloping terrain and from a moment balancearound the axle of the rear sprocket, the following relations are obtained.

T2 + T4 = T3 cos θ′ti − P sin θ′

ti − W sin β (6.5)

W cos β = T3 sin θ′ti + P cos θ′

ti (6.6)

Q = T1Rr = T3Rr (6.7)

Thence, the driving or braking force T1, the effective driving or braking force T4 and theground reaction P can be calculated as:

T1 = T3 (6.8)

T4 = T3

cos θ′ti

− W sin(θ′ti + β)

cos θ′ti

− T2 (6.9)

P = W cos β

cos θ′ti

− T3 tan θ′ti (6.10)

Here, under the conditions of id = ib = 0, the above equations are satisfied for T2 = T4 = 0,T3 = W sin(θ′

t0 + β), θ′ti = θ′

t0 and P = W cos(θ′t0 + β).

Next, let us consider equilibrium in the energy domain. The effective input energy E1

supplied by the driving or braking torque acting on the rear sprocket must equal the totaloutput energies i.e. the sum of the energy components comprise of the compaction energyE2 that is required to make a rut under the flexible track belt, the slippage energy E3 thatdevelops in the shear deformation of the soil beneath the flexible track belt, the effectivedriving or braking force energy E4 and the potential energy E5. This yields the followingbalance equation.

E1 = E2 + E3 + E4 + E5 (6.11)

212 Terramechanics

In addition, the energy expenditure or consumption per unit of time can be worked out fromthe vehicle speed V and the circumferential speed V ′ of the flexible track belt as follows:

(1) During driving action

E1 = T1V ′ = T3V ′ = T3V

1 − id(6.12)

E2 = T2V ′(1 − id) = T2V (6.13)

E3 = T3

(1 − 1 − id

cos θ′ti

)V ′ + WV ′(1 − id) tan θ′

ti cos β

= T3

(1

1 − id− 1

cos θ′ti

)V + WV tan θ′

ti cos β (6.14)

E4 = T4V ′(1 − id) = T4V (6.15)

E5 = WV ′(1 − id) sin β = WV sin β (6.16)

(2) During braking action

E1 = T1V ′ = T3V ′ = T3V (1 + ib) (6.17)

E2 = T2V ′

1 + ib= T2V (6.18)

E3 = T3(1 + ib) cos θ′

ti − 1

(1 + ib) cos θ′ti

V ′ + WV ′ 1

1 + ibtan θ′

ti cos β

= T3

{(1 + ib) − 1

cos θ′ti

}V + WV tan θ′

ti cos β (6.19)

E4 = T4V ′

1 + ib= T4V (6.20)

E5 = WV ′

1 + ibsin β = WV sin β (6.21)

6.2 FLEXIBLE DEFORMATION OF A TRACK BELT

As shown in Figure 6.4, the flexible deformation characteristics of the track belt of a flexiblytracked vehicle depend mainly on the track belt tension T0, the ground reaction Fp, and theshear resistance Fs acting on the interface between the soil and the track belt. For a sectionCCm of the flexible track belt as shown in the diagram, one can set-up some force balanceequations between the forces T0 and Fs acting in the longitudinal direction to the row ofroad rollers and between Fp acting in the normal direction to the row of road rollers. Thisprocess yields the following relationships:

T0 + Fs = T0 + �T0 (6.22)


Figure 6.4. Forces acting on a flexible track belt.

Fp = T0 tan δ = T0ds′

oi(X )

dX(6.23)

In these equations Fs is the integral of the shear resistance of the soil acting along the Xaxis i.e. along the line of the row of road rollers from the coordinate X to Xm correspondingto the points C to Cm, and it equals the increase of track belt tension. Fp is computableas the integral of the normal stresses that act on the track belt as one passes from X toXm. The angle of deflection of the track belt at an arbitrary point C is designated as δ. Thepoint Cm is taken to be situated at the position where the amount of deflection of the trackbelt takes on a maximum value and where the angle δ takes a value of zero. The parameters′

0i(X ) is the static amount of sinkage of the flexible track belt at a distance X from thebottom-dead-center B of the front-idler where it contacts with the main straight part of thetrack belt.

The force Fp = F(X ) can be approximated as the integral of the normal contact pressurepi(X ) from X to Xm assuming a rigid track belt as follows:

F(X ) = B∫ Xm

Xpi(X ) dX (6.24)

where B is the width of the track belt.The force T0 = T0(X ) can be calculated as the sum of the initial track belt tension H0 and

the thrust Hm(X ) i.e. the integral of the shear resistance τi(X ) of soil acting on the interfacebetween the track belt and the terrain over the range X = 0 to X = X , as follows:

T0(X ) = H0 + Hm(X ) = H0 + B∫ X

0τi(X ) dX (6.25)

Then, the static amount of sinkage s′0i(X ) can be calculated by integrating Eq. (6.23). The

result is:

s′0i(X ) =

∫ Xm

X

F(X )

T0(X )dX + s′

0i(Xm) − {s′

0i(D) − s′0i(0)

} Xm − X

D(6.26)

214 Terramechanics

Further, the distributions of the normal stress pi(X ) that acts under a flexible track belt canbe sub-analysed as follows:

For the situation where 0 ≤ s′0i(X ) ≤ H

p′i(X ) = k1

{s′

0i(X )}n1 (6.27)

For the situation where s′0i(X ) > H

p′i(X ) = k1H n1 + k2

{s′

0i(X ) − H}n2 (6.28)

In the unloading phase that occurs after the passage of a road roller, the distribution of thenormal contact pressure pi(X ) at a distance X where s′

0i(X ) becomes less than the peakvalue of static amount of sinkage s′

0i(Xt) at the coordinate Xt of each road roller [1] can besub-analysed as follows:

For the situation where 0 ≤ s′0i(Xt) ≤ H

p′i(X ) = k1

{s′

0i(Xt)}n1 − k3

{s′

0i(Xt) − s′0i(X )

}n3 (6.29)

For the situation where s′0i(Xt) > H

p′i(X ) = k1H n1 + k2

{s′

0i(Xt) − H}n2 − k4

{s′

0i(Xt) − s′0i(X )

}n4 (6.30)

where the normal stress at the front-idler p′fi and at the rear sprocket p′

ri are given as p′i(0)

and p′i(D) respectively.

Additionally, as mentioned in the previous Sections 5.2.6 and 5.3.6, the static amountof sinkage s′

0i(X ) should be calculated by taking into account the fact that the integratedvalue of p′

i(X ) acting on the main straight part of the track belt must always be equal to thevalue of the real ground reaction P′ = P − Pf . Thence, the total amount of sinkage of thefront-idler s′

fi and the rear sprocket s′ri in the case of 0 ≤ s′

f 0i ≤ s′r0i can be expressed as:

s′fi = (s′

f 0i + s′fs) cos θ′

ti (6.31)

s′ri = (s′

r0i + s′rs) cos θ′

ti (6.32)

θ′ti = sin−1

(s′

ri − s′fi

D

)(6.33)

Under the conditions where s′f 0i > s′

r0i > 0, the total amount of sinkage of the rear sprockets′

ri following after the passage of the front-idler should be greater than the total amount ofsinkage of the front-idler s′

fi. Thus, s′ri given in the above Eq. (6.32) needs to be modified

as follows:

s′ri = (s′

f 0i + s′rs) cos θ′

ti (6.34)

Here, s′f 0i = s′

0i(0) and s′r0i = s′

0i(D), and s′fs and s′

rs are the amounts of slip sinkage of thefront-idler and the rear sprocket respectively.


The amount of sinkage s′i(Xt) at each contact point Xt of the front-idler, the road rollers

and the rear sprocket to the main straight part of the track belt are required to lie on thesame straight line, so that the following identity can be established.

s′i(Xt) = s′

fi + (s′ri − s′

fi)Xt

D(6.35)

Further, the total amount of sinkage s′i(X ) of the flexible track belt at a distance X can be

calculated as follows:

s′i(X ) = s′

0i(X ) + s′i(Xt) − s′

0i(Xt)

+ [{s′

i(Xt+1) − s′0i(Xt+1)

}− {s′

i(Xt) − s′0i(Xt)

}] X − Xt

Rp(6.36)

where Rp is the spacing interval of each road roller.The thrust or drag T3 can be calculated, as will be discussed later, as the sum of the

components Tmb, T ′mb, T ′′

mb, Tms, T ′ms, T ′′

ms that act on the main contact part of the track belt,the components Tfb, Tfs that act on the front-idler, and the elements Trb, T ′

rb, Trs and T ′rs that

act on the rear sprocket. Also, the compaction resistance T2 can be calculated for a rut depths′

ri > H in the manner already mentioned in conjunction with the previous Eq. (5.103). Thus,

T2 = 2B

[∫ H

0k1zn1 dz +

∫ s′ri

H

{k1H n1 + k2(z − H )n2

}dz

]

= 2k1B

n1 + 1H n1+1 + 2k1BH n1 (s′

ri − H ) + 2k2B

n2 + 1(s′

ri − H )n2+1 (6.37)

6.3 SIMULATION ANALYSIS

Figure 6.5 shows a flow chart that can be used to calculate the traffic performance ofa flexible tracked vehicle ascending a soft sloping terrain of slope angle β during drivingaction and also for descending down a terrain during braking action.

As input data to the flow chart the complete dimensions of the vehicle, its speed andthe shape of the terrain are required i.e. the vehicle weight W , the track width B, the trackcontact length D, the radius of the front-idler Rf , the radius of the rear sprocket Rr , theradius of the road roller Rm, the grouser height H , the eccentricity of the center of gravityG of the vehicle e, the height of the center of gravity G from the bottom surface of thetrack belt hg , the distance of the point of application of tractive effort F from the centerline of the bulldozer ld , the height of the point F from the bottom surface of the track belthd , the initial track belt tension H0, the slope angle of the terrain β, as well as the vehiclespeed V and the circumferential speed of the track belt V ′. Then as a second form of inputdata, the terrain-track system constants k1, n1, k2, n2, k3, n3, k4, n4 obtained from track-model-plate loading and unloading test results, and the other terrain-track system constantsmc, mf , a, k0, n0 are required. The constants fm, fs, jm and K1, K2, jm as well as c0, c1, c2

that can be obtained from the plate traction, untraction and reciprocal traction test resultsas mentioned previously in Chapter 4 are also needed. After these factors are available,the contact pressure distributions pf 0, pr0 and p0(X ) for a tracked vehicle at rest can be

216 Terramechanics

Figure 6.5. Flow chart to analyse the performance of flexible tracked vehicles during driving andbraking action.

calculated (assuming a rigid track belt) for the static amounts of sinkage of front-idler sf 0

and rear sprocket sr0, and the linear distribution of static amount of sinkage s0(X ) underthe rigid track belt, and the angle of inclination of vehicle θt0 in each case of (1) ∼ (5) asmentioned in the previous Section 5.1.2, and it should be iteratively calculated until theamount of sinkage s0(X ) is uniquely determined.


Next, the slip ratio id during driving action or the skid ib during braking action canbe calculated by recursion. The system of calculation may be divided into three differentcalculation streams on the basis of the value of the eccentricity of ground reaction ei (justcalculated) for ranges of ei < −1/6, −1/6 ≤ ei ≤ 1/6 and ei > 1/6.

The real ground reaction acting on the main part of the track belt P′ can be calculated fromthe apparent ground reaction P given in the previous Eq. (5.54) by subtracting the groundreaction acting on the contact part of the front-idler Pf given in the previous Eq. (5.117).The linear distribution of the static amount of sinkage sf 0i, sr0i, s0i(X ) given in the previousEq. (5.46) and the angle of inclination of the vehicle θt0i given in the previous Eq. (5.79)can be calculated from the real ground reaction P′, and these should be calculated until thecontact pressure distribution pi(X ) given in the previous Eqs. (5.47) and (5.48) is uniquelydetermined. The amounts of slip sinkage of the front-idler sfs and the rear sprocket srs

can be calculated from the previous Eqs. (5.99) and (5.101) respectively. Then, the totalamounts of sinkage of the front-idler sfi and the rear sprocket sri can be calculated for thesituation where 0 ≤ sf 0i ≤ sr0i from the previous Eqs. (5.97) and (5.98) respectively, and forthe situation where sf 0i > sr0i > 0 from the previous Eqs. (5.97) and (5.104). Thence, thedistribution of the amount of sinkage si(X ) and the angle of inclination of the vehicle θti

can be determined from the previous Eqs. (5.102) and (5.52) respectively.As a next step in the calculation process, the static amounts of sinkage of the front-idler

s′f 0i and the rear sprocket s′

r0i of the flexible track belt and the distribution of the staticamount of sinkage s′

0i(X ) can be worked out from Eq. (6.26).Then, the contact pressure p′

fi at the front-idler and p′ri at the rear sprocket, and the

contact pressure distribution p′i(X ) acting on the flexible track belt can be calculated from

Eqs. (6.27) ∼ (6.30).The total amounts of sinkage s′

fi at the front-idler and s′ri at the rear sprocket can be

calculated in Eqs. (6.31), (6.32) or (6.34) from the amounts of slip sinkage s′fs at the front-

idler and s′rs at the rear sprocket as mentioned later. Also, the angle of inclination of the

vehicle θ′ti can be calculated from Eq. (6.33). Finally, the distribution of the total amount

of sinkage s′i(X ) can be calculated from Eq. (6.36).

The thrust or drag T3 can be calculated as the sum of Tmb and Tms i.e. the integral of theshear resistance given as a function of the distribution of normal contact pressure p′

i(X )which acts on the interface between the soil and the base area and the side parts of thegrousers in the main part of the flexible track belt respectively, Tfb and Tfs i.e. the integralof the shear resistance of the soil acting on the base area and the side parts of grousers inthe contact parts of the front-idler respectively, and Trb, Trs i.e. the integral of the shearresistance of the soil acting on the base area and the side parts of the grousers in the contactparts of the rear sprocket, respectively.

The driving or braking force T1, the compaction resistance T2 given in Eq. (6.37), theeffective driving or braking force T4 given in Eq. (6.9), the ground reaction P given inEq. (6.10), the total amount of sinkage s′

fi, s′ri and the angle of inclination of the vehicle

θ′ti should be iteratively calculated until the thrust or drag T3 is uniquely determined. After

that, the relations between the distribution of track tension T0(X ), each energy componentE1, E2, E3, E4, and E5, the distribution of amount of slippage j(X ) under track belt, thedistribution of normal stress p′

i(X ) and shear resistance τ ′i (X ), the distribution of amount

of deflection of the track belt and the slip ratio i can be calculated and these relationshipscan be drawn graphically. Further, the various functions T1, T4 − i, s′

fi, s′ri − i, θ′

ti − i,

218 Terramechanics

ei − i and the tractive or braking power efficiency Ed , Eb − i can be plotted graphically byuse of a micro-computer. Finally, the optimum effective driving or braking force T4opt atthe optimum slip ratio or skid iopt and the maximum effective driving or braking force atthe slip ratio or skid im can be simultaneously determined.

6.3.1 At driving state

In this section, several methods for the calculation of a number of factors i.e. the amountof slip sinkage s′

fs, s′rs during driving action, the total amounts of sinkage of the front-idler

s′fi and of the rear sprocket s′

ri and the thrust T3 developed under the flexible track belt, willbe presented in detail. These factors relate to the flow chart of Figure 6.5.

The amount of slip sinkage s′fs at the front-idler as shown in Eq. (6.31) can be calculated

by substituting the contact pressure distribution pf (θm) acting on the contact part of thefront-idler and the amount of slippage of soil jfs during driving action into the previousEq. (4.8) as follows:

s′fs = c0

M∑m=1

{pf (θm) cos θm

}c1

{(m

Mjfs

)c2

−(

m − 1

Mjfs

)c2}

(6.38)

where

θm = θf

(1 − m

M

)

θf = cos−1

(cos θ′

ti − s′fi

Rf + H

)− θ′

ti

pf (θm) = k1{s(θm)

}n1 (0 ≤ s(θm) ≤ H )

= k1H n1 + k2{s(θm) − H

}n2 (s(θm) > H )

s(θm) = (Rf + H ){cos(θm + θ′

ti) − cos(θf + θ′ti)}

cos(θm + θ′ti)

jfs = (Rf + H ) sin θfid

1 − id(6.39)

The amount of slip sinkage s′rs at the rear sprocket as expressed in Eq. (6.32) can be

calculated by substituting the contact pressure distribution p′i(X ) acting on the main part of

the flexible track belt and the amount of slippage of soil js during driving action into theprevious Eq. (4.8) as follows:

s′rs = s′

fs + c0

N∑n=1

{p′

i

(n

ND

)}c1 {( n

Njs

)c2

−(

n − 1

Njs

)c2}

(6.40)

where

js = i′dD

1 − i′d


i′d = 1 − 1 − idcos θ′

ti

As a next step in the process, a force balance equation involving the thrust T3 during drivingaction can be set up as follows:

T3 = Tmb + Tms + Tfb + Tfs + Trb + Trs (6.41)

In this equation, Tmb and Tms are components of thrust that act along the base area and theside parts of the grousers of the main part of the flexible track belt, respectively. Thesethrust components can be calculated as follows:

Tmb = 2B∫ D

0

{mc + mf p′

i(X )} [

1 − exp{−a(jw + jf + i′dX )

}]dX (6.42)

Tms = 4H∫ D

0

{mc + mf

p′i(X )

πcot−1

(H

B

)}

× [1 − exp

{−a( jw + jf + i′dX )}]

dX (6.43)

jf = (Rf + H )[θf − (1 − id)

{sin(θf + θ′

ti) − sin θ′ti

}](6.44)

The elements Tfb and Tfs are those components of thrust that act on the base area and theside parts of the grousers of the contact part of the front-idler respectively. The magnitudeof these elements can be calculated as follows:

Tfb = 2B(Rf + H )∫ θf

0

∥∥{mc + mf pf (θ)}

× [1 − exp

{−ajf (θ)}]

cos θ − pf (θ) sin θ∥∥ dθ (6.45)

Tfs = 4H (Rf + H )∫ θf

0

{mc + mf

pf (θ)

πcot−1

(H

B

)}

× [1 − exp

{−ajf (θ)}]

cos θ dθ (6.46)

jf (θ) = (Rf + H )[(θf − θ) − (1 − id)

{sin(θf + θ′

ti) − sin(θ + θ′ti)}]+ jw (6.47)

The further factors Trb and Trs i.e. the components of thrust acting on the base area andthe side parts of the grousers of the contact part of the rear sprocket, respectively can becalculated as follows:

Trb = 2B(Rr + H )∫ θr

0

∥∥{mc + mf · pr(δ)} [

1 − exp{−ajr(δ)

}]× cos(θ′

ti − δ) + pr(δ) sin(θ′ti − δ)

∥∥ dδ (6.48)

Trs = 4H (Rr + H )∫ θr

0

{mc + mf

pr(δ)

πcot−1

(H

B

)}

× [1 − exp

{−ajr(δ)}]

cos(θ′ti − δ) dδ (6.49)

jr(δ) = (Rr + H ){(θ′

ti − δ) − (1 − id) (sin θ′ti − sin δ)

}+ i′dD + jf + jw (6.50)

220 Terramechanics

In the above equations, jw is the amount of slippage that occurs along the sloping terraindue to the component of the vehicle weight W sin(θ′

ti + β) that acts on the main contactpart of the flexible track belt. On a sloping terrain, the amount of slippage jw is distributeduniformly along the terrain-track interface at a slip value id = 0 or at a skid value ib = 0.The value of jw can be calculated from the following force balance equation which can beset-up between the vehicle weight component and the integral of the shear resistance i.e.

W sin(θ′ti + β) = ±2B

∫ D

0

{mc + mf p′

i(X )} {

1 − exp(±ajw)}

dX (6.51)

and therefore the slippage

jw = ±1

alog

[1 ∓ W sin(θ′

ti + β)

2B∫ D

0

{mc + mf p′

i(X )}

dX

](6.52)

is obtained. Here, jw takes on a positive value for values θ′ti +β > 0 in relation to the upper

sign and takes on a negative value for θ′ti + β < 0 in relation to the lower sign.

Further, when the eccentricity ei of the ground reaction lies outside the middle-third ofthe main contact part of the flexible track belt during driving action, the thrusts T ′

mb andT ′

ms acting on the base area and side parts of the grousers of the main contact part of thetrack belt and the thrusts T ′

rb and T ′rs acting on the base area and side parts of the grousers

of the contact part of the rear sprocket force for ei > 1/6, and the thrusts T ′′mb and T ′′

ms actingon the base area and side parts of the grousers of the main contact part of the track belt forei < −1/6 can be calculated in the same way as has already been mentioned in Chapter 5.These procedures are shown in the following equations:

For s′f 0i < 0 < H < s′

r0i

T ′mb = 2B

∫ D

D−L

{mc + mf p′

i(X )}× ∥∥1 − exp

[−a{jw + i′d(X − D + L)

}]∥∥ dX

L = s′r0i

s′r0i − s′

f 0i

D (6.53)

T ′ms = 4H

∫ D

D−L

{mc + mf

p′i(X )

πcot−1

(H

B

)}

× ∥∥1 − exp[−a

{jw + i′d(X − D + L)

}]∥∥ dX (6.54)

The values of T ′rb and T ′

rs can be calculated by substituting the amount of slippage jr(δ)given in the next equation into Eqs. (6.48) and (6.49).

jr(δ) = (Rr + H ){(θ′

ti − δ) − (1 − id)(sin θ′ti − sin δ)

}+ i′dL + jw (6.55)


For s′f 0i > H > 0 > s′

r0i

T ′mb = 2B

∫ LL

0

{mc + mf p′

i(X )} [

1 − exp{−a(jw + jf + i′dX )

}]dX

LL = s′f 0i

s′f 0i + s′

r0i

D (6.56)

T ′′ms = 4H

∫ LL

0

{mc + mf

p′i(X )

πcot−1

(H

B

)}[1− exp

{−a(jw + jf + i′dX )}]

dX (6.57)

The ground reaction P can be calculated from Eqs. (6.5), (6.6) and (6.37), (6.41) as follows:

P = 1

cos θ′ti

(W cos β − T3 sin θ′ti) (6.58)

Following this, an optimum effective driving force T4opt can be defined as the effectivedriving force T4 at the optimum slip ratio iopt when the effective driving energy E4 takeson a maximum value for a constant circumferential speed V ′ of the flexible track belt.Likewise, a tractive power efficiency Ed can be defined as follows:

Ed = (1 − id)T4

T1(6.59)

6.3.2 At braking state

In the flow chart of Figure 6.5, several procedures for calculating the amounts of slipsinkage s′

fs, s′rs that occur during braking action and for determining the total amounts of

sinkage of front-idler s′fi and the rear sprocket s′

ri and the drag T3 that develop under theflexible track belt are outlined in detail.

The amount of slip sinkage s′fs at the front-idler as expressed in Eq. (6.31) can be calculated

by substituting the amount of slippage jfs as computed in the next equation, into Eq. (6.38).

jfs = −(Rf + H )ib sin θf (6.60)

The amount of slip sinkage s′ri at the rear sprocket as expressed in Eq. (6.32) can be calculated

by substituting the amount of slippage js, as calculated in the next equation, into Eq. (6.40).

js = −i′bD

i′b = (1 + ib) cos θ′ti − 1 (6.61)

As a next step, the drag T3 that develops during braking action can be calculated fromEq. (6.41). For the situation where jf + jw ≥ 0, the respective drags Tmb and Tms acting onthe base area and the side parts of the grousers of the main part of the flexible track belt

222 Terramechanics

can be calculated using the following equations – as shown in the previous Section 5.3.3,

Tmb = 2B∫ DD

0

{τp − k0(jq − jm)n0

}dX

− 2B∫ D

DD

{mc + mf · p′

i(X )} [

1 − exp{−a(jq − jm)

}]dX (6.62)

Tms = 4H∫ DD

0

{τp − k0(jq − jm)n0

}dX

− 4H∫ D

DD

{mc + mf

p′i(X )

πcot−1

(H

B

)} [1 − exp

{−a(jq − jm)}]

dX (6.63)

where

jm = i′b1 + i′b

X + jf + jw

jf = (Rf + H )[θf − 1

1 + ib

{sin(θf + θ′

ti) − sin θ′ti

}](6.64)

and DD can be given for jf + jw > 0 as follows:

DD = −(jf + jw − jq)(

1 + 1

i′b

)(6.65)

Further, the drags Tfb and Tfs acting on the base area and the side parts of the grousers onthe contact part of the front-idler can be calculated by substituting the following expressionfor the amount of slippage jf (θ) into the previous Eqs. (5.160) and (5.161), respectively.

jf (θ) = (Rf + H )[

(θf − θ) − 1

1 + ib

{sin(θf + θ′

ti) − sin(θ + θ′ti)}]

(6.66)

The drags Trb and Trs acting on the base area and the side parts of the grousers on thecontact part of the rear sprocket can be also calculated by substituting the following amountof slippage jr(δ) into the previous Eqs. (5.169) and (5.170), respectively.

jr(δ) = (Rr + H ){

(θ′ti − δ) − 1

1 + ib(sin θ′

ti − sin δ)}

+ i′bD

1 + i′b+ jf + jw (6.67)

Additionally, when the eccentricity ei of the ground reaction falls outside the middle-thirdof the main contact part of the flexible track belt during braking action, the drags T ′

mb andT ′

ms that act on the base area and the side parts of the grousers of the main contact part ofthe track belt and the drags T ′

rb and T ′rs that act on the base area and the side parts of the

grousers of the contact part of the rear sprocket for ey > 1/6, and the thrusts T ′′y and T ′′

yacting on the base area and the side parts of the grousers of the main contact part of thetrack belt for ei < 1/6 can be calculated in the same way as has been previously discussedin Chapter 5. The process is as shown in the following equations.


For s′f 0i < 0 < H < s′

r0i

T ′mb = −2B

∫ D

D−L

{mc + mf p′

i(X )}

×∥∥∥∥1 − exp

[a

{jw + i′b

1 + i′b(X − D + L)

}]∥∥∥∥ dX (6.68)

T ′ms = −4H

∫ D

D−L

{mc + mf

p′i(X )

πcot−1

(H

B

)}

×∥∥∥∥1 − exp

[a

{jw + i′b

1 + i′b(X − D + L)

}]∥∥∥∥ dX (6.69)

The values of T ′rb and T ′

rs can be calculated by substituting the amount of slippage jr(δ),given in the next equation, into Eqs. (6.67) and (6.68) respectively.

jr (δ) = (Rr + H ){

(θ′ti − δ) − 1

1 + ib(sin θ′

ti − sin δ)}

+ i′bL

1 + i′b+ jw (6.70)

For s′foi > H > 0 > s′

roi and jf + jw < 0

T ′′mb = −2B

∫ LL

0

{mc + mf p′

i(X )}

×∥∥∥∥1 − exp

[a

{jw + i′b

1 + i′b(X − D + L)

}]∥∥∥∥ dX (6.71)

T ′′ms = −4H

∫ LL

0

{mc + mf

p′i(X )

πcot−1

(H

B

)}

×∥∥∥∥1 − exp

[a

{jw + i′b

1 + i′b(X − D + L)

}]∥∥∥∥ dX (6.72)

For jf + jw ≥ 0, it is necessary to calculate the values of T ′′mb and T ′′

ms by classifying theminto either of three cases i.e. 0 ≤ DD ≤ LL, LL < DD ≤ D or DD ≥ D. In these expressionsthe value of DD is as given in Eq. (6.63) – as mentioned in the Section 5.3.3.

Further, in the case where s′foi > H > 0 > s′

roi, it is necessary to carefully calculate the val-ues of T ′′

mb given in Eq. (6.56) and T ′′ms given in Eq. (6.57) during driving action and, likewise,

the values of T ′′mb given in Eq. (6.71) and T ′′

ms given in Eq. (6.72) during braking action. Thecalculation may be done in the same way as shown in the previous Eqs. (5.118) ∼ (5.121)and Eqs. (5.149) ∼ (5.152). In these calculations, specific consideration must be given tothe fact that the cohesive factor mc will apply on the track belt in the range of LL < X < D asa consequence of the touching of the rear sprocket following the passage of the front-idleron the terrain even if the normal contact pressure p′

i(X ) becomes zero.Next, the ground reaction P can be calculated from Eq. (6.58) in the same manner as

has been discussed for the driving state. The optimum effective braking force T4opt during

224 Terramechanics

braking action can be defined as the effective braking force T4 at the optimum skid ibopt

when the effective input energy |E1| takes a maximum value at a constant vehicle speed V .In addition, the braking power efficiency Eb can be defined as:

Eb = 1

1 + ib· T4

T1(6.73)

6.4 THEORY OF STEERING MOTION

Figures 6.6(a), (b), (c) give front, side-elevation and plan views of a tracked vehicle. Alsoshown are the primary dimension and the systems of forces that act on the vehicle whenit is running on a weak flat terrain with a turning motion. The dimension D is the contactlength of the track belt, B is the track width, H is the grouser height and Gp is the grouserpitch. The dimension C is the distance between the center-lines of the inner and the outertrack. The parameter rf is the radius of the front-idler. Likewise rr is the radius of the rearsprocket. The sinkages sfi, sfo and sri, sro are the amounts of sinkage of the front-idler andthe rear sprocket for the inner and outer tracks, respectively. W is the total vehicle weightwhilst Wi and Wo are the components of the vehicle weight that are distributed to the innerand to the outer track belts, respectively.

In terms of the various forces that act on the machine, T4lat is the lateral effective tractiveeffort and TL is an additional lateral force acting at a point F which may be transmitted from asecond connecting vehicle, depending on the direction of the effective tractive effort. From

Figure 6.6(a). Front view of a flexible tracked vehicle during turning motion showing principaldimensions and forces.


Figure 6.6(b). Side view (innertrack) of a flexible tracked vehicle during turning motion.

Figure 6.6(c). Plan (resultant slip velocity Vsi, Vso) view of a flexible tracked vehicle during turningmotion.

226 Terramechanics

Figure 6.6(a), the values of Wi and Wo can be calculated by use of the following equations:

Wi = W

{1

2− hg(tan θlat)

C

}+ (TL − T4lat)

{hd

C+ (tan θlat)

2

}−(

TL

2− T4lati

)tan θlat

(6.74)

Wo = W

{1

2− hg(tan θlat)

C

}− (TL − T4lat)

(hd

C− tan θlat

2

)−(

TL

2− T4lato

)tan θlat

(6.75)

where θlat is the angle of lateral inclination of vehicle.

θlat = sin−1{(sG0 − sGi)/C}

sGi = sfi + (sri − sfi)(0.5 + e) (6.76)

sG0 = sf 0 + (sr0 − sf 0)(0.5 + e)

Here, hg is the height of the center of gravity G of the vehicle measured from the bottomtrack belt, hd is the height of the application point of the resultant effective tractive effortT4R, and e is the eccentricity of the center of gravity G.

The factor eD is the amount of eccentricity measured from the center line of the vehicle.Also, Wi and Wo can be assumed to act on the same position Gi, Go for inner and outertrack as the position G.

As shown in Figure 6.6(b), the forces T1i and T1o are the driving forces transmittedfrom the torques Qi and Qo of the rear sprocket, which are applied on the track belt at thebottom-dead-center of the rear sprocket for the inner and the outer track belts, respectively.T2i and T2o are the locomotion resistances i.e. the compaction resistances acting in front ofthe inner and outer track belts at the depths zi and zo, which can be calculated from eachrut depth, sri and sro [2].

The forces T3i and T3o are the thrusts developed along the track belt under the interfacebetween the terrain and the grousers of the inner and outer track belts. These can becalculated as the integral of the shear resistance of the soil. Usually, the driving forceT1i(o) can be assumed to be the same as the thrust T3i(o), which depends on the shearingcharacteristics of the terrain.

T4loni and T4lono are the effective tractive forces acting on the inner and outer tracks. Thesecan be calculated from a force balance [3] as shown in the following expression:

T4loni(o) = T3i(o)

cos θti(o)− Wi(o) tan θti(o) − T2i(o) (6.77)

where θti(o) is the angle of longitudinal inclination of the inner and the outer track.The longitudinal effective tractive effort T4lon acting in the longitudinal direction of

the vehicle can be expressed as the sum of each effective tractive force T4loni and T4lono,as follows:

T4lon = T4loni + T4lono (6.78)

The lateral effective tractive effort T4lat acting in the lateral direction of the vehicle is givenas the integral of the shear resistance τi(o)lat(X ) that develops along the inner and outer track


belts, as follows:

T4lat = T4lati + T4lato

= B∫ D

0

{τilat(X ) + τolat(X )

}dX (6.79)

where X is the distance from the front part of the track belt. Theoretically, T4lat should takea value of zero because the turning moment resistance is only developed by the differenceof the effective tractive effort of inner and outer track, but it has some negligibly small valuedue to experimental errors in determining the terrain-track system constants in the lateraldirection of the track plate. In these calculations, the centrifugal force is not consideredbecause of its negligibly small value at low vehicle speeds. The angle δ of the resultanteffective tractive effort T4 is determined as:

δ = tan−1{(TL − T4lat)/T4lon}

(6.80)

The reactions Ppi and Ppo are the resultant normal ground reaction forces applied on theinner and the outer track, to which the amounts of eccentricity are eiD and eoD, respectively.H0i and H0o are the initial track tensions for the inner and the outer track belts.

As depicted in Figure 6.6(c), F is the point of application of the resultant effective tractiveeffort T4 on the vehicle. This is composed of a longitudinal component T4lon, and a lateralone T4lat . There is also an additional lateral force TL, for which the height hd is the distancemeasured from the bottom-dead-center of the rear sprocket and ld is the distance from thecenter line of the vehicle. Additionally, T4loni, T4lati and T4lono, T4lato can be assumed toact at the same position Fi and Fo for the inner and outer tracks as the position F on thelongitudinal plane.

The point P is the turning center of the tracked vehicle, measured from the center pointO of the tracked vehicle, i.e. R is the turning radius of the vehicle and Y is the deviationfrom the lateral center line of the vehicle. The elements Mi, Mo are the turning resistancemoments acting around the point Pi and Po of the inner and outer track. The deviations ofthese, from the center point Oi and Oo are Yi = eiD and Yo = eoD for the inner and outertracks, respectively.

The longitudinal effective tractive effort T4ion can be also derived from the followingmoment balance equation:

RT4lon + (ld − Y )(TL − T4lat)

=(

R − C

2

)(T3i

cos θti− Wi tan θti − T2i

)

+(

R + C

2

)(T3o

cos θto− Wo tan θto − T2o

)− Mi − Mo

=(

R − C

2

)T4loni +

(R + C

2

)T4lono − Mi − Mo (6.81)

Y = (e1 + e2) D/2 (6.82)

228 Terramechanics

Substituting Eq. (6.78) into the above equation, we get:

T4lono − T4loni = 2{Mi + M0 + (ld − Y )(TL − T4lat)

}/C (6.83)

When δ is zero, the difference between T4lono and T4loni can be calculated as 2(Mi + Mo)/Cusing Eq. (6.83) at TL = T4lat ≈ 0. When the additional lateral force TL becomes large, thedifference between T4lono and T4loni should be increased.

The resultant effective tractive effort T4R can be calculated as follows:

T4R = {(T4lon)

2 + (TL − T4lat)2}1/2

(6.84)

The parameter β is the slip angle of the center of gravity of the vehicle and βi and βo arethe slip angles of the inner and outer tracks, respectively. The value of these parameter isgiven by:

βi = tan−1{Y/(R − C/2)}

(6.85)

β0 = tan−1{Y/(R + C/2)}

(6.86)

β = tan−1{(Y − eD)/R}

(6.87)

The compaction resistance T2i(o) can be computed from the following expression:

T2i(o) = B∫ sri(o)

ok1sn1 ds (6.88)

6.4.1 Thrust and steering ratio

For the inner and outer tracks, the longitudinal slip velocity Vsi(o)lon and the longitudinalamount of slippage ji(o)ion(X ) at a distance X from the front part of the inner and outertracks can be calculated through use of the following two equations:

Vsi(o)lon = rri(o)ωi(o) − Vi(o) (6.89)

ji(o)lon(X ) =∫ t

o(rri(o)ωi(o) − Vi(o)) dt = ii(o)X (6.90)

where t is the movement-time of the track belt i.e. X /rri(o)ωi(o) from the front part to a pointX , and ii(o) is the slip ratio of the inner and outer tracks as will be discussed later.

The longitudinal shear resistance of the soil that develops under the inner and outer trackbelts τi(o)lon(X ) at points Ni and No may be calculated as:

τi(o)lon(X ) = (mclon + pi(o)(X )mflon)[1 − exp

{−alonji(o)lon(X )}]

(6.91)

Thence, the main thrust of the inner and outer tracks T3i(o) can be calculated as:

T3i(o) = B∫ D

0τi(o)lon(X ) dX (6.92)

When the circumferential speed of the rear sprocket of inner track and outer track is set tobe rriωi and rroωo, a steering ratio ε may be defined as follows:

ε = rroωo/rriωi (6.93)


Again, another steering ratio ε′ can be defined as in the following equation, where the speedof the inner and the outer tracks are set to be Vi and Vo at slip angles βi and βo, respectively:

ε′ = Vo cos βo/Vi cos βi (6.94)

In this case, the slip ratios of the inner and outer tracks ii, io are expressed as:

ii = 1 − Vi cos βi/rriωi (6.95)

io = 1 − Vo cos βo/rroωo (6.96)

when both the track belts are in the driving state.Substituting the above equations into Eq. (6.93), the following relationship can be

derived.

ε′ = ε(1 − io)/(1 − ii) (6.97)

The turning speed of the tracked vehicle V at the center of gravity G and the running speedof the inner and outer tracks Vi and Vo may be calculated as follows:

V = ω√

R2 + (Y − eD)2 (6.98)

Vi = rriωi(1 − ii)/cos βi = ω√

(R − C/2)2 + Y 2 (6.99)

Vo = rroωo(1 − io)/cos βo = ω√

(R + C/2)2 + Y 2 (6.100)

ω = rriωi(1 − ii)/(R − C/2) = rroωo(1 − io)/(R + C/2) (6.101)

where ω is the steering angular velocity of the tracked vehicle.Eliminating the steering angular velocity ω, the turning radius of the tracked vehicle R

can be determined as:

R = C{rroωo(1 − io) + rriωi(1 − ii)

}2{rroωo(1 − io) − rriωi(1 − ii)

} (6.102)

6.4.2 Amount of slippage in turning motion

Calculation of the lateral slip velocity between a track and a soil and the amount of lateralslippage of soil under a track belt in turning motion is required to determine the turningresistance moment of the inner and outer tracks.

Figure 6.7 shows the resultant slip velocity Vsi(o) whose components are rri(o)ωi(o) in thelongitudinal direction and ωPNi and ωPN0 in the tangential direction. As a consequence,the lateral slip velocity Vsi(o)lat(X ) of the inner and outer tracks at arbitrary points Ni and No

may be given, cf. also Figure 6.6(c), as the lateral component of the resultant slip velocity.

Vsi(o)lat(X ) = ω sin α√

(R ± C/2)2 + (D/2 − X + Y )2

= ω(D/2 − X + Y )

= Vi(o)D/2 − X + Y√(R ± C/2)2 + Y 2

(6.103)

230 Terramechanics

Figure 6.7. Lateral amount of slippage jilat , jolat and shear resistance τilat , τolat for inner and outertrack.

Thence, the lateral amount of slippage ji(o)lat(X ) can be calculated as:

ji(o)lat(X ) =∫ t

0Vsi(o)lat(X ) dt

= Vi(o)√(R ± C/2)2 + Y 2

∫ t

o(D/2 − X + Y ) dt

= Vi(o)

rri(o)ωi(o)

√(R ± C/2)2 + Y 2

∫ X

o(D/2 − X + Y ) dX

= 1 − ii(o)

cos βi(o)

√(R ± C/2)2 + Y 2

(D/2 + Y − X /2)X

= 1 − ii(o)

R ± C/2(D/2 + Y − X /2)X (6.104)

where t is the time of travel of a grouser from the front part of the track belt to arbitrarypoints Ni and No under the inner and outer track. This time is given as X /rri(o)ωi(o). As aconsequence, it is evident that the lateral amount of slippage ji(o)lat(X ) takes on a value ofzero at X = 0 and D + 2Y . It also takes a maximum value at X = D/2 + Y for both innerand outer tracks.


6.4.3 Turning resistance moment

The lateral shear resistance τi(o)lat(X ) that develops along a track belt at points Ni and No,under an inner and outer track, can be calculated through use of the following expression.

dji(o)lat(X )/dX ≥ 0

τi(o)lat(X ) = {mclat + pi(o)(X )mflat

} [1 − exp

{−alat ji(o)lat(X )}]

(6.105)

dji(o)lat(X )/dX < 0 jq ≤ ji(o)lat(X ) ≤ jp

τi(o)lat(X ) = τp − k3(jp − j)n3 (6.106)

dji(o)lat(X )/dX < 0 ji(o)lat(X ) < jq

τi(o)lat(X ) = − {m′clat + mflat

′pi(o)(X )} [

1 − exp{−a′

lat

{jq − ji(o)lat(X )

}}](6.107)

where jp = ji(o)lat(D/2 + Y ), tp = ti(o)lat(D/2 + Y ), jq = jp − (tp/k3)1/n3 , and pi(o)(X ) is thenormal pressure distribution under the inner and outer track belt.

The parameters mclat , mflat and alat , k3, n3, and m′clat , m′

flat and a′lat are the lateral terrain-

track system constants which were measured – including the bulldozing resistance of thetrack model plate – via the track plate traction test. It is noted here that the reaction ofthe additional lateral force TL is automatically included in this distribution of lateral shearresistance.

Following this, the turning resistance moments Mi and Mo that are exerted around theturning points Pi and Po of the inner and outer track, can be calculated – including theamount of sinkage of the track plate – as follows:

Mi(o) = B∫ D

oτi(o)lat(X )(D/2 − X + Y ) dX (6.108)

The total turning resistance moment M is given by the following equation:

M = Mi + M0 (6.109)

The energy equilibrium equation for the straight forward motion of a tracked vehicle hasalready been presented [4]. For machines in turning motion, the input energy E1i(o) suppliedby the driving torque can be equated to the sum of: the compaction energy E2i(o) requiredto make a rut under the track belt, the slippage energy E3i(o) required to develop a thrustalong the bottom of the track belt, the effective tractive effort energy E4i(o), and the turningmoment energy E5i(o) for the inner track and the outer track. That is,

E1i(o) = E2i(o) + E3i(o) + E4i(o) + E5i(o) (6.110)

where

E1i(o) = T1i(o)Vi(o) cos βi(o)/(1 − ii(o))

E2i(o) = T2i(o)Vi(o) cos βi(o)

232 Terramechanics

E3i(o) = T3i(o){1/(1 − ii(0)) − 1/(cos θti(o))

}Vi(o) cos βi(o)

+ Wi(0)Vi(o) cos βi(o) tan θti(o)

E4i(o) = T4loni(o)Vi(o) cos βi(o) + {TL/2 − T4lati(o)

}(ld − ei(o)D)ω

E5i(o) = ωMi(o)

Then the total input energy E1 of the vehicle and the total output energy of the compactionenergy E2, the slippage energy E3, the effective tractive effort energy E4 and the turningmoment energy E5 of the vehicle can be given as in the equations:

E1 = E2 + E3 + E4 + E5 (6.111)

E1 = E1i + E1o (6.112)

E2 = E2i + E2o (6.113)

E3 = E3i + E3o (6.114)

E4 = E4i + E4o (6.115)

E5 = E5i + E5o (6.116)

This energy equilibrium equation can also be proved theoretically by using the force andmoment balance equations as mentioned above. An optimum resultant effective tractiveeffort T4Ropt may be defined as the resultant effective tractive effort at the optimum com-bination of slip ratio iiopt of the inner track and ioopt of the outer track which takes themaximum value of the effective tractive effort energy E4max. A tractive power efficiency Ed

may be developed as follows:

Ed = E4/E1 (6.117)

6.4.4 Flow chart

As illustrated in Figure 6.8, the necessary input information for a simulation analysis of aflexibly tracked machine operating on a terrain includes the weight W , the contact lengthD of the inner and the outer tracks, the track width B of the inner and the outer tracks, thecentral distance C between the inner and the outer tracks, the radius rf of the front-idler, theradius rr of the rear sprocket, the radius of the track roller rm, the number of track rollers N ,the grouser height H , the eccentricity of the center of gravity e of the vehicle, the height hg

of the center of gravity of the vehicle, the distance ld between the central axis of the vehicleand the application point, the height hd of the application point of the total effective tractiveeffort, the initial track belt tension H0. Following the input of this primary geometrical andmass data, as a next step the terrain-track system constants k1, k2 and n1, n2 from the plateloading and unloading test, mclon, mclat ; mflon, mflat ; alon, alat from the plate traction test,and c0lon, c0lat ; c1lon, c1lat ; c2lon, c2lat from plate slip sinkage test need to be read in as data.

At rest, the static amount of sinkage sfi = sfo and sri = sro, the linear distribution of staticamount of sinkage si(X ) of inner track which equals so(X ) of outer track at the distance Xfrom the contact point of front-idler on the main part of track belt, the angle of longitudinalinclination of inner and outer track θti = θto, the contact pressure pfi = pfo at the front-idler and pri = pro at the rear sprocket, the nonlinear normal pressure distribution pi(X ) of


DATA W, B, D, C, rf, rr, rm, N, H, e, hg, ld, hd, Ho

READ k1, n1, k2, n2, k3, n3, mclon, mflon, alon, c0lon, c1lon, c2lon,mclat, mflat, alat, mclat, mflat, alat c0lat, c1lat, c2lat

' ' '

sfi =sfo, sri =sro, si(X) =so(X), uti = uto, pfi =pfo, pri =pro, pi(X) =po(X),ei= eo

At rest

|si(X) – si'(X)|� ε'

N

rrivi, rrovo,ε

ii

Wi, Wo

ei < -1/6

ppi, pfi, pi(X)sfi, sri, si(X), uti

| si - si |� ε2'

ppi, pfi, pi(X)' ' '

sfi, sri, si(X),' ' '

T1i, T2i, T3i, T4i, ppi, uti, ei, Mi, Vi, T4lon'

| T3i - T3i < ε3' |

-1/6�ei�1/6 ei > 1/6

eo < -1/6 -1/60� eo�1/6 eo > 1/6

io

|T4lon - T4lon | < ε4'

'

T4, TL, T4lon, T4lat, d, R, Y, V, b, v, ε, ε', ulat, M, E1, E2, E3, E4, E5, Ed, ii, T1i, T2i, T3i, T4loni, ppi, Vi, bi, ei, si(X) uti', Mi, E1i, E2i, E3i, E4i, E51, jilon(X), jilat(X), pi'(X), τilon'(X), τilat'(X), Toi(X), io, T1o, T2o,T3o, T4lono,T4lato, ppo,Vo, bo, uo, So'(X), uto, Mo, E1o, E2o, E3o, E4o, E5o, jolon(X), jolat(X), po'(X), τolon'(X),τolat'(X),To

T4opt, iiopt, ioopt, T4max, iim, iom END/STOP

ppi, pfi, pi(X)sfi, sri, si(X),uti

ppi, pfi, pi(X)sfi, sri, si(X),uti

| si - si � ε2' | | si - si |� ε2'

ppi, pfi, pi(X)' ' '

sfi, sri, si(X),' ' 'ppi, pfi, pi(X)' ' '

sfi, sri, si(X),' ' '

ppo, pf0, po(X)sfo, sro, so(X),uto

| so - so � ε2' |

ppo, pfo, po(X)' ' '

sfo, sro, so(X),

ppo, pfo, po(X)sfo, sro, so(X), uti

| so – so � ε2'|

ppo, pfo, pi(X)' ' '

sfo, sro, so(X),' ' '

ppo, pfo, po(X)sfo, sro, so(X), uto

| so – so � ε2'

ppo, pfo, po(X)' ' '

sfo, sro, so(X),' ' '

T1o, T2o, T3o, T4o, ppo, uto', eo, Mo, Vio, T4lon

| T3o - T3o | < ε3'

N

N

N

N

N NN

N

N

At Driving state

|

| ulat- ulat | < ε5

Figure 6.8. Flow chart of flexible track.

the inner track which equals po(X ) of the outer track, and the eccentricity ei = eo of theresultant force Ppi = Ppo for the assumed rigid track belt can be iteratively calculated untilthe distribution of static amount of sinkage si(X ) = so(X ) is determined.

For a given angular velocity of the rear sprocket ωi for the inner track and ωo for the outertrack, and for a steering ratio ε, both the tractive performances of the inner and the outertrack during driving action can be calculated for each combination of slip ratio ii and io.

234 Terramechanics

First of all, the tractive performance of the inner track is calculated for the slip ratio ii ofthe inner track, assuming that the distributed vehicle weight Wi and Wo equals half thevehicle’s weight i.e. W /2 respectively.

For the above calculated eccentricity ei, the resultant normal force Ppi, the contact pres-sure pfi, pri and pi(X ), the total amount of sinkage sfi, sri and si(X ) including the amount ofslip sinkage, and the angle θti can be iteratively calculated (depending on the three kinds ofprocedure calculations streams which are divided according to the value of eccentricity ei)until the distribution of the total amount of sinkage si(X ) is determined. In order to trans-form the above results that have been calculated for an assumed rigid track belt into thoseof an actual flexible track belt, the normal contact pressure distribution and the distributionof the total amount of sinkage need to be changed to p′

fi, p′ri, and p′

i(X ) and s′fi, s′

ri, and s′i(X )

for the flexible track belt considering the initial track belt tension H0i. This process has beendiscussed in a previous paper [5]. Then, the driving force T1i, the compaction resistance T2i,the thrust T3i from Eq. (6.92), the effective tractive force T4i from Eq. (6.77), the angle θti,the eccentricity ei of the normal resultant force Ppi, the turning resistance moment Mi fromEq. (6.108), the running speed Vi from Eq. (6.99), and the longitudinal effective tractiveeffort of the vehicle T4lon from Eq. (6.78) can be iteratively calculated until the thrust T3i isdetermined.

As a next step, the tractive performance of the outer track is calculated for the slip ratio ioof the outer track assuming that the distributed vehicle weight Wo equals W /2. For the abovecalculated eccentricity eo = ei, the ground reaction Ppo, the contact pressure pfo, pro andpo(X ), the total amount of sinkage sfo, sro and so(X ) including the amount of slip sinkage,and the angle θto can be repeatedly calculated (depending on the three kinds of procedurecalculations streams which are divided according to the value of eccentricity ei) until thedistribution of the total amount of sinkage so(X ) is determined. In order to transform theabove results calculated for the assumed rigid track belt to the actual flexible track belt,the normal contact pressure distribution and the distribution of the total amount of sinkageshould be changed to p′

fo, p′ro, and p′

o(X ) and s′fo, s′

ro, and s′o(X ) for the flexible track belt

considering the initial track belt tension H0o. Then, the driving force T1o, the compactionresistance T2o, the thrust T3o from Eq. (6.92), the effective tractive force T4o from Eq. (6.77),the angle θto, the eccentricity eo of the ground reaction Ppo, the turning resistance momentMo from Eq. (6.108), the running speed Vo from Eq. (6.100), and the longitudinal effectivetractive effort of the vehicle T4lon from Eq. (6.78) can be iteratively calculated until thethrust T3o is determined.

Thereafter, the actual slip ratio io of the outer track for the given slip ratio ii of the innertrack can be calculated repeatedly by using the two division method until the longitudinaleffective tractive effort of the vehicle T4lon from Eq. (6.83) is determined. After that, the realdistributions of vehicle weight to the inner and the outer track Wi and Wo can be calculatedrecursively from Eqs. (6.74) and (6.75) until the real angle of lateral inclination of thevehicle θlat is determined.

Following this, the tractive performance of the vehicle i.e. the resultant effective tractiveeffort T4R from Eq. (6.84) which is composed of the longitudinal effective tractive effortT4lon from Eq. (6.78) and the lateral effective tractive effort T4lat from Eq. (6.79), and theangle δ of the vehicle, the position of the turning center of the turning radius of the vehicleR from Eq. (6.102) and Y from Eq. (6.82), the running speed V and the angle β, the steeringratio ε from Eq. (6.93) and ε′ from Eq. (6.94), the angle of lateral inclination of the vehicle


θlat , the total turning resistance moment M from Eq. (6.109), and the total amount of theinput energy E1, the compaction energy E2, the slippage energy E3, the effective tractiveeffort energy E4 and the turning moment energy E5 from Eqs. (6.112) ∼ (6.116), and thetractive power efficiency Ed from Eq. (6.117) can be determined for each combination ofslip ratio ii and io.

In addition, the tractive performances of the inner track and the outer track i.e. the slipratio ii(o), the driving force T1i(o), the compaction resistance T2i(o), the thrust T3i(o), the effect-ive tractive force T4i(o), the running speed Vi(o) and the angle βi(o), the eccentricity ei(o) ofthe resultant normal force Ppi(o), the distribution of the total amount of sinkage si(o)(X ), theangle of longitudinal inclination θti(o), the turning resistance moment Mi(o), the distributionof the longitudinal amount of slippage ji(o)lon(X ) and the lateral amount of slippage ji(o)lat(X ),the normal contact pressure distribution pi(o)(X ), the shear resistance distribution τi(o)(X ),the distribution of track tension T0i(o)(X ), the input energy E1i(o), the compaction energyE2i(o), the slippage energy E3i(o), the effective tractive effort energy E4i(o) can be determinedin detail. Finally, the optimum effective tractive effort T4Ropt and the optimum combinationof slip ratio iiopt of the inner track and ioopt of the outer track, can be determined fromEq. (6.117).

6.5 SOME EXPERIMENTAL STUDY RESULTS

In general, the soft terrain under a flexible track belt (which is the typical undercarriagesystem on many pieces of construction machinery such as bulldozer or tractors) is takento failure or is transformed into the plastic state. This occurs, because the contact pressureof the track plate reaches several times that of the average contact pressure and exceedsthe bearing capacity of the terrain when the axle load of a road roller is supported byonly a few track plates located just under the road roller. It is generally expected thatthe terrain under the flexible track belt moves into a state of active earth pressure whenthe amount of deflection of track belt becomes convex to the terrain, while it moves intoa state of passive earth pressure for a concave deflection of track belt to the terrain [6]. Inthis section, various traffic performance aspects of a flexible tracked vehicle, especially theaspect of characteristic contact pressure distribution under the flexible track belt during self-propelling action and whilst operating under traction, are presented.

6.5.1 During self-propelling operation

From the literature, it is well known that the contact pressure distribution under a flexibletrack belt has a wavy sinusoidal distribution having peak values under the individual roadrollers.

Rowland [7] reported that the maximum contact pressure measured at a depth of 23 cmwas 1.2 to 2.0 times larger than the average contact pressure of a flexible tracked machineequipped with several types of undercarriage, road rollers and suspension systems duringself-propelling states on loose accumulated sandy terrain and soft terrain composed ofclayey soil or muskeg etc. Cleare [8] measured the maximum contact pressure at depth of30 cm of a soft silty terrain as being 1.3 times larger than the average contact pressure of aflexible tracked vehicle. Again, Fujii et al. [9] measured the distribution of contact pressure

236 Terramechanics

Figure 6.9. Relationship between contact pressure σz under flexible track belt at depth of 14.5 cmand passive time t (self propelling state) [5].

under the flexible track belt of a tractor of weight 113 kN, contact length of track belt 220 cmand track width 40 cm when the tractor was self-propelling on a convex sandy terrain ofbulk density 16.3 kN/m3 and water content 19 ∼ 24%. Figure 6.9 shows an example ofcontact pressure distribution under the flexible track belt. The pressures were measuredby use of an earth pressure cell buried at a depth of 14.5 cm. In this case, the maximumcontact pressure dropped down to about 0.63 times that of the average contact pressure of64 kPa – possibly because the measured point might have deviated from the center line ofthe running line of the tractor.

Sofiyan et al. [10] measured directly the distribution of contact pressure under the flexibletrack belt of a tractor running on a sandy terrain by use of strain gauges attached directlyto track links. They reported that the maximum contact pressure reached 3.0 ∼ 3.5 timeslarger that the average contact pressure. Thus, it becomes clear that the maximum value,period or wavy pattern of the contact pressure depends on the soil properties and the surfaceroughness of the terrain, the size or number of road rollers, the mechanism of connectionand the suspension system, the track tension between the pin joints of the track links, theshape of track plate, the structure of track belt and so on.

There are not many available experimental studies measuring the behaviour of soil par-ticles under a flexible track belt, but Yong et al. [11] measured the elliptical trajectories ofsoil particles under a rigid wheel during driving action at various slip ratios. They suggestedthat the relative amounts of movement between a flexible track belt and a terrain becomevery large and the soil particles under the flexible track belt move very significantly in thehorizontal and vertical direction especially immediately under the road rollers. Experimen-tally, it was observed that the soil particles under a flexible track belt during self-propellingstate move forward in front of the road rollers and move backward to their rear sides dueto the comparatively small track tension [12]. The shear resistance acting on the flexibletrack belt takes on a positive or negative value in correspondence to the alternative relativeamounts of slippage between the track plate and the terrain under the road rollers.


Figure 6.10. Mechanism of rolling frictional resistance of road roller on flexible track belt.

Next, some information on the rolling frictional resistance Tr of a road roller rotating onthe track link of a flexible track belt will be presented. Figure 6.10 shows a mechanismillustrating the dynamic rolling movement of each road roller of axle load Wi rotating ontwo track links separated at the distance of one track plate length l [13]. If one assumesthat the track plates, with grousers mounted on the track links, are operating on a hardterrain, each track link will rotate mutually around the link pin of radius r for a centralangle of αi during one pass of the road roller. For an amount of vertical movement δi of theaxle of road roller during one pass of the track plate, the increment of potential energy ofthe road roller equates to Wiδi. Then, an energy equilibrium equation for one pass of theroad roller on the track link of length of l can be established as:

Trl = µRln∑

i=1

Wi +n∑

i=1

Wiδi + 2rm∑

i=1

αifi

where Tr is the rolling frictional resistance, µR is the coefficient of rolling resistance, fi isthe frictional force between pin and bush of track link and m and n are, respectively, thenumber of links and track plates for one contact length of track belt.

Thence, the rolling frictional resistance of road roller on a flexible track belt Tr can beworked out as follows:

Tr = µR

n∑i=1

Wi + 1

l

(n∑

i=1

Wiδi + 2rm∑

i=1

αifi

)(6.118)

Further, this rolling frictional resistance Tr increases the driving torque of the rear sprocketas an internal resistance, especially for the self-propelling state of vehicle when not somuch track belt tension develops.

Bekker [14, 15] investigated the relations that exist between the amount of sinkage ofa flexible track belt and the number of road rollers. He found that the amount of sinkage wasdecreased 34% by doubling the number of road rollers and that the amount of sinkagedecreased rapidly when the number of rollers increased from 2 to 5. But it then tended toa constant value as the number increased from 5 to 9.

238 Terramechanics

Figure 6.11. Contact pressure distribution of bulldozer under traction [19].

6.5.2 During tractive operations

The distribution of contact pressure under a flexible track belt (as installed on a typical bull-dozer or tractor under tractive operations) changes from a uniform wavy distribution to a trap-ezoidal or triangular wavy one with increases in effective tractive effort. The re-distributionof the contact pressure is caused by the occurrence of a moment around the center of gravityof the vehicle since the effective tractive effort is applied at some height to the rear part ofthe bulldozer. Wills [16] and Torii [17] have confirmed that the contact pressure distributionof a tractor under traction shows some wavy triangular patterns towards to the rear end ofthe track belt. Wong [18] observed that the distribution of the contact pressure acting onflexible track belt under traction at various slip ratios shows some wavy distribution withseveral peak values just under the road rollers. He also reported that the ratio of maximumcontact pressure to the average one takes on a value of 9.8 for sandy terrain, 2.9 for muskegterrain and 8.3 for snow covered terrain. Ito et al. [19] measured the contact pressure dis-tribution under a bulldozer of weight 349 kN running on a Kanto loam terrain of watercontent 108.7%, bulk density 13.4 kN/m3 and cone index 1068 kPa.

Figure 6.11 shows the contact pressure distribution in self-propelled action and underdriving action developing two kinds of effective tractive effort of 55.9 kN and 131.3 kNi.e. the contact pressure at the rear end of track belt in the wavy triangular distribution andthe eccentricity of ground reaction tended to increase with increase in the effective tractiveeffort.

Also, Bekker [20] investigated the variation in the contact pressure distribution actingunder a flexible track belt for a tractor of weight 30 kN running at various vehicle speeds andvarious degrees of effective tractive effort on a weak terrain having various water contents.

He determined that the maximum contact pressure occurred at the rear end of the flexibletrack belt and that it increased with increasing tractive effort and with a decrease in thewater content of the terrain.

6.6 ANALYTICAL EXAMPLE

In the published literature, there have been a number of methods of theoretical analysispresented that relate to the problem of predicting the tractive performances of flexible


Table 6.1. Dimensions of bulldozer running on soft terrain.

Vehicle weight W (kN) 50Track width B (cm) 100Contact length D (cm) 320Average contact pressure pm (kPa) 7.81Radius of front-idler Rf (cm) 50Radius of rear sprocket Rt (cm) 50Grouser height H (cm) 12Grouser pitch Gp (cm) 36Interval of road roller Rp (cm) 40Radius of road roller Rm (cm) 8Eccentricity of center of gravity of vehicle e −0.02Height of center of gravity of vehicle hg (cm) 100Distance of application point of effective driving or braking lb (cm) 300

force from central axis of vehicle ld (cm) 300Height of application point of effective driving hb (cm) 50

or braking force hd (cm) 50Initial track belt tension Ho (kN) 9.8Circumferential speed of track belt (during driving state) V ′ (cm/s) 100Vehicle speed (during braking state) V (cm/s) 100

tracked vehicle on a soft terrain. For example, we have Yong’s energy [21] and FEM [22]analyses concerning the interaction between the structure of a track belt and a terrain.

Likewise we have Garber’s numerical analysis [23] which considers the strength of theground, the track belt tension and the distribution of contact pressure, Wong’s simulationanalysis [24] and Oida’s study [25] which may be used to calculate theoretically the thrustof a flexible track belt from the shear deformation properties of soil on an unit track plate.

In this section, several analytical examples relating to the trafficability of several flexibletracked vehicles running during driving and braking action on a silty loam terrain, a decom-posed weathered granite sandy terrain and a snow covered terrain with considerations ofthe size effect of the track model plate and the initial track belt tension will be presented.

6.6.1 Silty loam terrain

(1) Trafficability of a bulldozer running on soft terrainThe complex tractive performance during driving action and the complex braking perfor-mance during braking action of a flexible tracked vehicle of bulldozer running on a softterrain are developed here through use of a rigorous mathematical simulation method. Thespecifications of the vehicle that will be used in these studies are given in Table 6.1.

The structure of the track belt is that of a flexible rubber track belt equipped with equi-lateral trapezoidal grousers of trim angle α = π/6 rad, contact length L = 4 cm and grouserpitch of Gp = 36 cm as shown in the previous Figure 4.16. The vehicle is specified to berunning on a flat silty loam terrain of slope angle of β = 0 rad. To allow for the size effectof track model plate, the terrain-track system constants as shown in Table 6.2 have alreadybeen modified by substituting a size ratio of N = 8 into the previous Eqs. (4.36) ∼ (4.45).

(a) In the driving stateBy substituting the above mentioned vehicle specifications and terrain-track system con-stants as input data into the flow chart of the previous Figure 6.5, the variations of driving

240 Terramechanics

Table 6.2. Terrain-track system constants for bulldozer running on soft terrain.

Track plate loading and unloading testp ≤ 17.8 (kPa) k1 = 1.254 n1 = 1.068p > 17.8 (kPa) k2 = 5.513 n2 = 0.895

k3 = k4 = 30.04 n3 = n4 = 0.632

Track plate traction testmc = 3.362 (kPa) mf = 0.311 a = 0.078 (l/cm)

Track plate slip sinkage testc0 = 0.692 c1 = 0.584 c2 = 0.478

Figure 6.12. Relationship between driving force T1 effective driving force T4 and slip ratio id (duringdriving action).

force T1, the effective driving force T4, the amount of sinkage of the front-idler s′fi and of

the rear sprocket s′ri, the angle of inclination of the vehicle θ′

ti, the eccentricity of the groundreaction ei, each energy component E1 ∼ E4, and the tractive power efficiency Ed , with theslip ratio id during driving action can be calculated.

In looking at the results, Figure 6.12 is a plot of the obtained relations between T1, T4 andid . The driving force T1 increases initially and then tends to a constant value with increasingvalues of id . In contrast the effective driving force T4 moves quite rapidly to a maximumvalue of T4max = 16.3 kN at a slip of id = 10%. After that it decreases gradually to takea value of zero at id = 23%. The vehicle can not develop any traction force at a slip ratio ofid greater than 23%.

Figure 6.13 plots the relations between the sinkages s′fi, s′

ri and the slip ratio id . From thediagram it is evident that s′

ri increases with increasing values of id accompanied by increasingamounts of slip sinkage. As a consequence, the angle of inclination of the vehicle θ′

ti, whichis determined from s′

fi and s′ri increases gradually with increasing id as shown in Figure 6.14.

The eccentricity ei decreases with id for small magnitudes of slip ratio and takes a minimumvalue of−0.193 at id = 51%.After this it increases again with id . The eccentricity ei becomesless than −1/6 for the range of id = 29 ∼ 67%.


Figure 6.13. Relationship between amount of sinkage, of front-idler S ′fi rear sprocket S ′

ri and slip ratioid (during driving action).

Figure 6.14. Relationship between angle of inclination of vehicle θ′ti eccentricity of ground reaction

ei and slip ratio id (during driving action).

Figure 6.15 plots the relationships between the various energy components and the slip ratio.That is the effective input energy E1, compaction energy E2, slippage energy E3, effectivedriving force energy E4 are plotted as a function of the slip ratio id . The effective inputenergy E1 moves to a constant value with increasing values of id , but the compaction energyE2 follows a Hump type curve with a maximum value. The slippage energy E3 increasesalmost linearly with id . The effective driving force energy E4 takes a maximum value of1471 kN cm/s at an optimum slip ratio of idopt = 10%. After this it decreases gradually to aminimum value at id = 71%, then increases again.

242 Terramechanics

Figure 6.15. Relationship between energy elements E1, E2, E3, E4 and slip ratio id (during drivingaction).

Figure 6.16. Relationship between tractive power efficiency Ed and slip ratio id (during drivingaction).

Figure 6.16 plots the relationship between tractive power efficiency Ed and slip ratio id .The tractive power efficiency Ed takes a maximum value of 57.0% at id = 3% after that itdecreases rapidly with id to take on a minimum value of −86.9% at id = 68%. After thisminimum, it then increases again.

The conclusions here are, that when this bulldozer is running on this silty loam terrainduring driving action and operating at idopt = 10% under with maximum driving force


Figure 6.17. Contact pressure distribution under flexible track belt (during driving action).

Figure 6.18. Distribution of track belt tension T0 (during driving action).

energy, T1opt = 35.1 kN and T4opt = 16.3 kN can be developed at s′fi = 6.0 cm, s′

ri = 19.8 cm,θ′

ti = 0.043 rad, ei = −0.082 and Ed = 41.8%.The contact pressure distribution acting on the flexible track belt of this bulldozer varies

with the slip ratio id . As an example, Figure 6.17 shows the distribution of normal stressp′

i(X ) and shear resistance τ ′i (X ) at id = 10%. As shown in the diagram, there are several

stress concentrations just under the road rollers and the contact pressure between the roadrollers decreases due to the deflection of the flexible track belt. That is, the dynamic loadacts repeatedly on the terrain. The pressure p′

i(X ) tends to increase toward the forward partof the track belt due to the negative eccentricity of the ground reaction. As will be discussedlater, the amount of deflection of the front part of flexible track belt increases due to therelatively small track belt tension so that the amplitude of the contact pressure increases atthe front part of the track belt. In contrast, τ ′

i (X ) shows the same wavy distribution, but,at the front part of the track belt, the shear resistance τ ′

i (X ) does not develop to any greatdegree because of the small amount of slippage.

Figure 6.18 shows the distributions of track belt tension T0 around the track belt at id = 10,20 and 30%. The track belt tension increases toward the rear part of the flexible track beltwith id , while it equals the initial track belt tension H0 at the part of the front-idler. At thebase part of the rear sprocket, T0 reaches 34.5 kN at id = 30%. Furthermore, the amountof deflection of the track belt corresponds to the distribution of the track belt tension T0.For instance, the track belt at the base of the rear sprocket is tensioned to an amount ofdeflection of 3.1 mm while the amount of deflection at the base of the front-idler is 7.1 mm.

244 Terramechanics

Figure 6.19. Relationship between braking force T1 effective braking force T4 and skid (duringbraking action).

Figure 6.20. Relationship between amount of sinkage of front-idler s′fi rear sprocket s′

ri and skid ib

(during braking action).

(b) In the braking stateIn the same way as has already been discussed, variations of braking force T1, effective brak-ing force T4, amount of sinkage of the front-idler s′

fi and the rear sprocket s′ri, angle of inclina-

tion of vehicle θ′ti, eccentricity of ground reaction ei, energy components E1 ∼ E4, braking

power efficiency Eb during braking action, can be calculated as functions of the skid ib.In terms of analysing the results, Figure 6.19 plots the relations between T1, T4 and the

skid ib. |T1| increases rapidly with the increasing |ib| for the lower ranges of skid, but it tendsgradually to a constant value for the upper ranges of skid. On the other hand, |T4| increasesgradually with increments in |ib|. It is always larger than |T1| because of the increasingcompaction resistance T2 that occurs with increasing |ib|.


Figure 6.21. Relationship between angle of inclination of vehicle θ′ti, eccentricity of ground reaction

et and skid ib (during braking action).

Figure 6.22. Relationship between energy elements E1, E2, E3, E4 and skid ib (during braking action).

Figure 6.20 plots the relations between the sinkages s′fi, s′

ri and the skid ib. The rear sprocketsinkage s′

ri increases gradually with increments in |ib| in company with increased amountsof slip sinkage. In contrast s′

fi takes on an almost constant value. As a consequence, the angleof inclination of the vehicle θ′

ti, which is determined from s′fi and s′

ri, increases parabolicallywith increases in |ib| as shown in Figure 6.21. The eccentricity ei increases almost linearlywith |ib| with a negative value to positive value transition at ib = −36%.

Figure 6.22 plots the relations between the various energy components and the skid. Thatis the effective input energy E1, the compaction energy E2, the slippage energy E3, the effec-tive braking force energy E4 are plotted as a function of the skid ib. |E1| decreases graduallyto zero at ib = −100% after taking on a maximum value of 3525 kN cm/s at ibopt = −19%.

246 Terramechanics

Figure 6.23. Contact pressure distribution under flexible track belt (during braking action).

Figure 6.24. Distribution of track belt tension T0 (during braking action).

The effective braking force energy |E4| increase parabolically while E2 and E3 increasesgradually and almost linearly with increasing values of |ib|.

In summary, when this bulldozer is running on this silty loam terrain and is operat-ing during braking action at ibopt = −19% under the maximum effective input energy,T1opt = −43.6 kN and T4opt = −60.5 kN can be developed at s′

fi = 4.1 cm, s′ri = 18.5 cm,

θ′ti = 0.045 rad, ei = −0.008 and Eb = 172%.The contact pressure distribution that acts on

the flexible track belt varies with the skid ib. To illustrate, Figure 6.23 shows the distribu-tions of the normal stress p′

i(X ) and the shear resistance τ ′i (X ) at ib = −20%. As shown in the

diagram, there are several stress concentrations just under the road rollers and the contactpressure between the road rollers decreases due to the deflection of the flexible track belt.As will be discussed later, the amount of deflection of the rear part of the flexible track beltincreases due to the relatively small track belt tension so that the amplitude of the contactpressure increases at the rear part of the track belt. On the other hand, τ ′

i (X ) shows a negativewavy distribution in the same way, but, at the front part of the track belt, the shear resistanceτ ′

i (X ) does not develop to any great extent because of the small amount of slippage.Figure 6.24 shows the distributions of track belt tension T0 around the track belt at

ib = −10, −20 and −30%. The track belt tension increases toward the front part of theflexible track belt with |ib|, whilst it equals the initial track belt tension H0 at the part of therear sprocket. At the base-part of the front-idler, T0 reaches 29.9 kN at ib = −30%. Further,the track belt at the part of the front-idler is tensioned to an amount of deflection of 1.6 mmwhile the amount of deflection at the base part of the rear sprocket is 3.3 mm.

(2) Size effect of vehicleSince the size effect of a track model plate on the tractive performances of a bulldozerrunning on a clayey terrain could be enormous, the problem of the size effect of vehicle


needs be considered seriously in any investigations of the trafficability of an actual flexibletracked vehicle as mentioned in the previous Section 4.3.4.

In the section that follows, several influences of the vehicle dimensions on the tractiveperformance are analysed when several bulldozers having various sizes of track belt andvehicle weight are running under traction on a flat silty loam terrain. These studies cancomprise a form of sensitivity analysis to machine scale.

Table 6.3 shows the vehicle dimensions of actual bulldozers and the terrain-track systemconstants calculated from the previous Eqs. (4.36) ∼ (4.45) for a size ratio of track modelplate of N = 4, 8, 12 and 16 respectively.

Using the flow chart to calculate the tractive performance of the flexible tracked vehicleas shown in Figure 6.5 and using this input data, the relationships between the effectivetractive effort T4, the amount of sinkage of the rear sprocket s′

ri, the angle of inclinationof the vehicle θ′

ti, the eccentricity of the ground reaction ei and the slip ratio id have beenmathematically simulated.

As to results, Figure 6.25 shows the computed effects of the size ratio N of the vehicle onthe relations between the coefficient of traction i.e. the effective tractive effort T4 dividedby the vehicle weight W and the slip ratio id . For a small bulldozer of N = 4, the coefficientof traction takes a maximum value of (T4/W )max = 0.824 at id = 38%. After that it becomeszero at id = 94%. In contrast, for a large bulldozer of N = 16, the coefficient of tractiontakes on a maximum value of (T4/W )max = 0.105 at id = 4% but the bulldozer can not workat id = 8% onwards.

From this work, it is clearly demonstrated that the coefficient of traction of a bulldozerrunning on a soft terrain decreases remarkably with increases of size and weight of thebulldozer.

For this case, the tractive performance of a bulldozer having the same average contactpressure of 7.81 kPa will drop down from a size ratio of N = 16 onward.

Figure 6.26 shows the effects of the size ratio N of the vehicle on the relations betweenthe relative amount of sinkage i.e. the amount of sinkage of the rear sprocket s′

ri dividedby the track width B and the slip ratio id . As the relative amount of sinkage s′

ri/B tends toincrease rapidly with increasing values of size ratio N for whole the range of slip ratio id ,the land locomotion resistance is considered to increase remarkably with the size of thetrack belt of the vehicle.

Figure 6.27 shows the relations between the angle of inclination of vehicle θ′ti and the slip

ratio id . The angle θ′ti increases with increases in the slip ratio id and with increases in the

size ratio N . This means that the vehicle will tend to incline remarkably with the increasesin the size of the track belt. The eccentricity of the ground reaction ei tends to change froma negative value to a positive one at some slip ratio id , as shown in Figure 6.28. The absolutevalue of the minimum eccentricity eimin tends to increase with increasing N . For a vehicleof size ratio N = 16, eccentricity ei reduces down to less than −1/6. As a consequencethe vehicle becomes unstable in the range of slip ratios of id = 23 ∼ 27%. Further, it isconfirmed that the tractive power efficiency at the optimum slip ratio also decreases withincreases in the size ratio N of the vehicle.

From the above simulation analytical results, it should be noticed that the coefficient oftraction and the tractive power efficiency at the optimum slip ratio of a bulldozer operatingon a weak silty loam terrain decrease remarkably, even if the average contact pressure is thesame, with increases in the size of vehicle due to a diminution of the terrain-track systemconstants sensitivity to the size effect of the track belt.

248 Terramechanics

Tabl

e6.

3.D

imen

sion

sof

bulld

ozer

runn

ing

onso

ftte

rrai

nan

dte

rrai

n-tr

ack

syst

emco

nsta

nts.

Veh

icle

size

ratio

N4

812

16V

ehic

lew

eigh

tW

(kN

)12

.550

.011

2.5

200.

0T

rack

wid

thB

(cm

)50

100

150

200

Con

tact

leng

thD

(cm

)16

032

048

064

0A

vera

geco

ntac

tpre

ssur

ep m

(kPa

)7.

817.

817.

817.

81R

adiu

sof

fron

t-id

ler

RI

(cm

)15

3045

60R

adiu

sof

rear

spro

cket

Rr

(cm

)15

3045

60In

terv

alof

road

rolle

rR

p(c

m)

2040

6080

Rad

ius

ofro

adro

ller

Rm

(cm

)4

812

16V

ehic

leH

eigh

tof

cent

reof

grav

ityh o

(cm

)30

6090

120

ofve

hicl

eD

imen

sion

sD

ista

nce

ofap

plic

atio

npo

int

I d(c

m)

150

300

450

600

ofef

fect

ive

driv

ing

forc

efr

omce

ntra

laxi

sof

vehi

cle

Hei

ghto

fap

plic

atio

npo

int

h d(c

m)

2550

7510

0of

effe

ctiv

edr

ivin

gfo

rce

Gro

user

heig

htH

(cm

)6

1218

24G

rous

erpi

tch

Gp

(cm

)18

3654

72In

itial

trac

kbe

ltte

nsio

nH

o(k

N)

2.45

9.80

22.0

539

.20

Ecc

entr

icity

ofce

nter

e0.

00of

grav

ityof

vehi

cle

Cir

cum

fere

ntia

lspe

edof

V′ (

cm/s

)10

0tr

ack

belt

k 1(N

/cm

n 1+2

)2.

871.

267.

76×

10−1

5.51

×10

−1L

oadi

ngte

stN

19.

41×

10−1

1.07

1.15

1.21

Terr

ain-

trac

kk 2

(N/c

mn 2

+2)

5.57

5.52

5.48

5.46

syst

emco

nsta

nts

n 28.

36×

10−1

8.95

×10

−19.

31×

10−1

9.58

×10

−1m

c(k

Pa)

4.09

3.36

3.00

2.76

mf

3.42

×10

−13.

11×

10−1

2.94

×10

−12.

82×

10−1

Tra

ctio

nte

st(1

/cm

)1.

39×

10−1

7.84

×10

−25.

61×

10−2

4.43

×10

−2c 0

(cm

2c1−c

2+1

/Nc 1

)2.

72×

10−1

6.93

×10

−11.

201.

76c 1

7.34

×10

−15.

86×

10−1

5.14

×10

−14.

68×

10−1

c 23.

80×

10−1

4.78

×10

−15.

46×

10−1

6.01

×10

−1


Figure 6.25. Relationship between coefficient of traction T4/W and slip ratio id for various size ratiosN of vehicle.

Figure 6.26. Relationship between relative amount of sinkage s′ri/B and slip ratio id for various size

ratio N of vehicle.

(3) Effect of initial track belt tensionIt is a well-established empirical fact that the tractive performance of a bulldozer running ona soft terrain depends to a very large degree on the flexibility of the track belt. Usually, thetrack belts on working machines are tensioned by use of an adjuster composed of a cylinderand a spring that is mounted on the axle of the front-idler. Typically, the initial track belttension is set up at more than 20% of the vehicle weight by use of a grease gun. However,under normal operations, the track belt tension reduces due to wear in the undercarriageparts of the bulldozer i.e. in the front-idler, track link, pin, bush, road roller, rear sprocketand so on. In general, the tractive effort of a bulldozer decreases with reductions in the

250 Terramechanics

Figure 6.27. Relationship between angle of inclination of vehicle θ′ti and slip ratio id for various size

ratio N of vehicle.

Figure 6.28. Relationship between eccentricity of ground reaction ei and slip ratio id for various sizeratio N of vehicle.

initial track belt tension because of an increasing land locomotion resistance with increasesin amounts of sinkage [26, 27].

In this section, several effects of the initial track belt tension on the optimum effectivetractive effort, the amount of sinkage of the rear sprocket, the eccentricity of the groundreaction, the angle of inclination of the vehicle and the tractive power efficiency at theoptimum slip ratio of a bulldozer of vehicle weight 150 kN are forecast through the use ofa rigorous mathematical simulation analysis program.

We assume the bulldozer to be simulated is equipped with a flexible track belt and tobe operating on a flat weak remolded silty loam terrain. The track belt is equipped withtrapezoidal rubber grousers of height 6 cm, pitch 18 cm, contact length 2.0 cm and baselength 8.9 cm. The terrain-track system constants that prevail between the track belt and asilty loam terrain of water content 30% and cone index 30 kPa are summarized in Table 6.4.


Table 6.4. Terrain-track system constants for track belt equipped with trapezoidal rubbergrousers.

Track plate loading and unloading testp ≤ 16.8 (kPa) k1 = 2.255 n1 = 1.120p > 16.8 (kPa) k2 = 6.669 n2 = 5.938 × 10−1

k3 = k4 = 30.03 n3 = n4 = 0.632

Track plate traction testmc = 3.626 (kPa) mf = 0.356 a = 0.148 (1/cm)

Track plate slip testc0 = 0.253 c1 = 0.751 c2 = 0.360

Table 6.5. Dimensions of tracked bulldozer equipped with trapezoidal rubber grousers runningon soft terrain.

Vehicle weight W (kN) 150Track width B (cm) 150Contact length D (cm) 320Average contact pressure pm (kPa) 15.6Radius of front-idler Rf (cm) 35Radius of rear sprocket Rt (cm) 35Grouser height H (cm) 6Grouser pitch Gp (cm) 18Interval of road roller Rp (cm) 40Eccentricity of center of gravity of vehicle e −0.05

00.05

Distance of application point M of effective driving force ld (cm) 310from central axis of vehicle

Height of application point of effective driving force hd (cm) 40Initial track belt tension H0 (kN) 0 ∼ 50Circumferential speed of track belt (during driving state) V ′ (cm/s) 100Height of center of gravity of vehicle hg (cm) 70

The dimensions and specifications of the bulldozer are shown in Table 6.5. For this systema mathematical simulation has been run for three values of eccentricity of the center ofgravity of the bulldozer e, namely e = −0.05, 0 and 0.05 and for ten kinds of initial trackbelt tension H0, namely H0 = 0, 2, 4, 6, 8, 10, 20, 30, 40 and 50 kN. Substituting thesevalues as input data into the flow chart of Figure 6.5, the effects of the initial track belttension on the tractive performances of the bulldozer may be analysed [28].

Figure 6.29 graphs the relationship between the optimum effective tractive effort T4opt

and the initial track belt tension H0 for various values of eccentricity of the center of gravityof vehicle e. It is especially evident that T4opt drops down with decreasing values of H0

from H0 = 10 kN for e = 0.00 or 0.05.Figure 6.30 shows the relationship between the amounts of sinkage of the rear sprocket s′

riat the optimum slip ratio and H0 for e = −0.05, 0.00 and 0.05. The rear sprocket sinkage s′

riincreases rapidly with decreasing values of H0 from H0 = 10 kN for each of the values of e.

Figure 6.31 shows the relationship between the angle of inclination of the vehicle θ′ti

at the optimum slip ratio and H0 for e values of −0.05, 0.00 and 0.05 respectively. The

252 Terramechanics

Figure 6.29. Relationship between optimum effective driving force T4opt and initial belt tension H0

for various kinds of eccentricity e of center of gravity of vehicle.

Figure 6.30. Relationship between amount of sinkage of rear sprocket s′ri and initial track belt tension


inclination θ′ti decreases gradually with decreasing values of H0 and then decreases rapidly

for H0 values less than 10 kN for each setting of e, where the vehicle can move with stability.Figure 6.32 shows the relationship between the eccentricity ei of the ground reaction at

the optimum slip ratio and H0 for various ranges of e. The eccentricity ei is almost a constantvalue independent of changes in H0. This applies for each value of e.

Figure 6.33 shows the relationship between the tractive power efficiency Ed at the opti-mum slip ratio and H0 for e = −0.05, 0.00 and 0.05. From this diagram, it is clear that Ed

decreases rapidly with decreasing values of H0 less than 10 kN for each value of e.


Figure 6.31. Relationship between angle of inclination of vehicle θ′ti and initial track belt tension H0


Figure 6.32. Relationship between eccentricity ei of ground reaction and initial track belt tension H0

for various e (id = idopt).

From the results of the above simulation analysis, it is evident that, for the various eccentric-ities of the center of gravity of the vehicle e = −0.05, 0.00 and 0.05, the optimum effectivetractive effort, the angle of inclination of vehicle and the tractive power efficiency at theoptimum slip ratio decreases rapidly with decreasing values of the initial track belt tension.However, the amount of sinkage of the rear sprocket at the optimum slip ratio increasesrapidly with decreasing values of the initial track belt tension.

In contrast to this situation, a rigid tracked vehicle equipped with a rubber track belthaving a great initial track belt tension can develop a larger optimum effective tractive

254 Terramechanics

Figure 6.33. Relationship between tractive power efficiency Ed and initial track belt tension H0 forvarious e (id = idopt).

Figure 6.34. Forces acting on a bulldozer running on decomposed granite sandy terrain.

effort and tractive power efficiency over and above those of a flexible tracked vehiclerunning on a soft terrain as discussed above.

In this case, it is necessary to control the initial track belt tension to be always larger than10 kN to maintain a sensible operation of a bulldozer running on a soft terrain.

6.6.2 Decomposed granite sandy terrain

In this section we will consider a simulation of the traffic performance of a flexibletracked vehicle, such as a bulldozer, running on a flat terrain composed of an accumulateddecomposed weathered granite sandy soil as already has been discussed in Section 4.3.2.

Figure 6.34 shows the system of forces acting on the bulldozer. The structure of the trackbelt is that of a flexible belt equipped with equilateral trapezoidal rubber grousers of base


Table 6.6. Dimensions of small bulldozer.

Vehicle weight W (kN) 50Track width B (cm) 25Contact length D (cm) 320Average contact pressure pm (kPa) 31.25Radius of frontidler Rf (cm) 50Radius of rear sprocket Rt (cm) 50Grouser height H (cm) 6.5Grouser pitch Gp (cm) 14.6Interval of road roller Rp (cm) 40Radius of road roller Rm (cm) 8Eccentricity of center of gravity of vehicle e −0.02Distance of application point M effective driving force from

central axis of vehicle ld (cm) 300Height of application point of effective driving force hd (cm) 60Initial track belt tension Ho (kN) 19.6Circumferential speed of track belt (during driving state) V ′ (cm/s) 100Height of center of gravity of vehicle hg (cm) 100

length L = 2 cm, grouser height H = 6.5 cm, grouser pitch Gp = 14.6 cm and trim angleα = π/6 rad as shown in the previous Figure 4.6. The terrain-track system constants thatoperate between the decomposed granite sandy soil of dry density 1.44 g/cm3 and a giventrack-model-plate of width B = 25 cm are k1 = 8.526 N/cmn1+2, n1 = 0.866, mc = 0 kPa,mf = 0.769, a = 0.244 cm−1, c0 = 1.588 cm2c1−c2+1/Nc1 , c1 = 0.075 and c2 = 0.240 asgiven in the previous Table 4.2.

Table 6.6 shows the specifications and dimensions of a small bulldozer of weight 50 kN,width of track belt 25 cm and contact length of track belt 320 cm. In this case, it is notnecessary to consider the size effect of the terrain-track system constants because the widthof the track belt is the same as that of the track-model-plate.

(1) At driving stateBy using the above terrain-track system constants and vehicle dimensions as the input datato the flow chart of Figure 6.5, the variations of driving force T1, effective driving forceT4, amounts of sinkage of the front-idler s′

fi and rear sprocket s′ri, angle of inclination of

the vehicle θ′ti, the eccentricity of the ground reaction ei, the various energy components

E1 ∼ E4 and the tractive power efficiency Ed can be calculated as a function of the slip ratioid for a vehicle under driving action.

Figure 6.35 portrays the relations between T1, T4 and id resulting from the simulationcalculations. The driving force T1 increases rapidly to a constant value with increasingvalues of id . The effective driving force T4 decreases gradually with id after peaking ata maximum value of T4max = 39.7 kN at id = 23%.

Figure 6.36 shows the relations between s′fi, s′

ri and id . The Figure shows both the sinkagess′

fi, s′ri increasing gradually with id . The sinkage s′

ri is always larger than s′fi because of the

increasing amount of slip sinkage at the rear sprocket. Thus, it can be seen that the angle ofinclination of the vehicle θ′

ti which may be calculated from s′fi and s′

ri increases parabolicallywith increasing values of slip ratio id . The phenomenon is illustrated in Figure 6.37. Asshown in the diagram, the eccentricity of the ground reaction ei has a negative value. Itdecreases gradually after it passes a maximum value of −0.0108 at id = 5%.

256 Terramechanics

Figure 6.35. Relationship between driving force T1 effective driving force T4 and slip ratio id .

Figure 6.36. Relationship between amount of sinkage of frontidler s′fi rear sprocket s′

ri and slip ratio id .

Figure 6.38 plots the relationships that exist between the various energy components andthe slip ratio id . Thus, Figure 6.38 shows the effective input energy E1, the compactionenergy E2, the slippage energy E3 and the effective tractive effort energy E4 as a functionof the slip ratio id . The effective input energy E1 increases quite rapidly and then tends toa constant value with id .

The energies E2 and E4 follow a hump-type curve having a maximum value. The com-ponent E3 increases almost linearly with increasing values of id . E4 peaks at a maximum



and slip ratio id (during driving action).

Figure 6.38. Relationship between energy elements E1, E2, E3, E4 and skid id (during driving action).

value of 3430 kNcm/s at id = 10% and after that it decreases almost linearly to zero atid = 100%.

Figure 6.39 shows the relations between the tractive power efficiency Ed and the slipratio id . Ed has a sharp peak with a maximum value of 90.4% at id = 2%. After that, itdecreases, essentially linearly, to zero at id = 100%.

In summary, when this small bulldozer is running on a decomposed weathered granitesandy terrain during driving action and is operating at idopt = 10% under maximum effectivetractive effort energy, T1opt = 41.1 kN and T4opt = 38.1 kN can be developed at s′

fi = 6.9 cm,s′

ri = 11.7 cm, θ′ti = 0.015 rad, ei = −0.0118 and Ed = 83.5%.

The contact pressure distribution acting on the flexible track belt of a small bulldozervaries with the slip ratio id . For example, Figure 6.40 shows the distribution of the normalstress p′

i(X ) and the shear resistance τ ′i (X ) for id = 10 ∼ 90% for the simulated system.

The normal stress p′i(X ) tends to increase toward the front part of the track belt due to the

negative value of ei. The amount of deflection of the track belt and the amplitude of the wavydistribution of the normal stress on the front part of the flexible track belt become largedue to the relatively small track belt tension. On the other hand, the shear resistance τ ′

i (X )also shows a wavy distribution, but the shear resistance does not develop significantly atthe front part of the track belt for id = 10% because the amount of slippage is very small.

258 Terramechanics

Figure 6.39. Relationship between tractive power efficiency Ed and slip ratio id (during drivingaction).

Figure 6.40. Contact pressure distribution (during driving action).

Figure 6.41. Distribution of track belt tension T0 (driving state).

Figure 6.41 shows the distribution of the track belt tension T0 around the track belt forid = 10 ∼ 90%. The track belt tension increases toward the rear part of the flexible trackbelt with id , whilst it shows an initial track belt tension of H0 = 19.6 kN at the base of thefront-idler. At the base of the rear sprocket, T0 reaches 36.2 kN at id = 10%. The track beltat the base of the rear sprocket is tensioned to provide a deflection of 0.43 mm while themagnitude of the deflection at the base of the front-idler is 1.96 mm for id = 10%.


Figure 6.42. Relationship between braking force T1 effective braking force T4 and skid ib.


ri and skid ib.

(2) At braking stateIn the same manner as has been already described, the variations of the braking force T1,the effective braking force T4, the amounts of sinkage of the front-idler s′

fi and rear sprockets′

ri, the angle of inclination of the vehicle θ′ti, the eccentricity of the ground reaction ei, the

various energy components E1 ∼ E4 and the braking power efficiency Eb as a function ofthe skid ib during braking action, can be calculated by use of the flow chart of Figure 6.5.In terms of quantitative results, Figure 6.42 shows the relations between T1, T4 and ib. Boththe values of |T1|, |T4| increase with increasing values of |ib| and tend to constant values.|T4| is always greater than |T1| for all values of skid.

Figure 6.43 shows the relations between the sinkages s′fi, s′

ri and ib. Both the values ofs′

fi and s′ri increase gradually with increasing values of |ib|. The rear sinkage s′

ri is always

260 Terramechanics


ei and skid ib (during braking action).

Figure 6.45. Relationship between energy elements E1, E2, E3, E4 and skid ib (during braking action).

larger than the front sinkage s′fi due to the amount of slip sinkage that occurs at the rear

sprocket.Figure 6.44 shows the relations between θ′

ti, ei and ib. The inclination θ′ti increases parabol-

ically with increasing values of |ib|. On the other hand, the eccentricity ei takes on a negativevalue which increases slightly after it takes a minimum value of −0.0409 at a skid ib = −8%.

Figure 6.45 shows the relations between the various energy elements – namely the effec-tive input energy E1, the compaction energy E2, the slippage energy E3, the effective brakingforce energy E4 and the skid ib. The element |E1| takes a maximum value of 3782 kNcm/sat an optimum skid ibopt of −13%. It then decreases gradually to zero at ib = −100%. Thecompaction energy E2 and the slippage energy E3 increases almost linearly with increasingvalues of |ib|, but |E4| increases parabolically and tends to a constant value with |ib|.


Figure 6.46. Contact pressure distribution (during braking action).

Figure 6.47. Distribution of track belt tension T0 (braking state).

In summary, when a simulated small bulldozer is in running operations on a decomposedweathered granite sandy soil during braking action and is operating at ib = −13% under amaximum effective input energy, T1opt = −43.5 kN and T4opt = −47.0 kN can be developedat s′

fi = 7.9 cm, s′ri = 13.0 cm, θ′

ti = 0.016 rad, ei = −0.041 and Eb = 124%.The contact pressure distribution acting on the flexible track belt of a small bulldozer

varies with the skid ib. For example, Figure 6.46 shows the simulated distributions of thenormal stress p′

i(X ) and the shear resistance τ ′i (X ) at ib = −10%.

The normal stress p′i(X ) tends to increase toward the front part of the track belt due to the

negative value of ei and there are several stress concentrations just under the track rollers.At the rear part of the track belt, both the amount of deflection of the track belt and theamplitude of the wavy distribution of the normal stress p′

i(X ) becomes large due to therelatively small track belt tension. On the other hand, the shear resistance τ ′

i (X ) shows anegative wavy distribution, but the shear resistance does not develop significantly at thefront part of the track belt because the amount of slippage is very small.

Figure 6.47 shows the distribution of the track belt tension T0 around the track beltat ib = −10%. The track belt tension increases toward the front part of the flexible trackbelt, whilst it shows an initial track belt tension of H0 = 19.6 kN at the base part of the rearsprocket. At the base of the front-idler, T0 reaches 36.8 kN at ib = −10%. The track belt at thebase part of the front-idler is tensioned such as to provide an amount of deflection of 0.1 mmwhile the amount of deflection at the part of the rear sprocket is 3.8 mm for ib = −10%.


The tractive performance during driving action and the braking performance during brakingaction of a flexible tracked over-snow vehicle of weight 40.0 kN running on a snow coveredterrain can be forecast through use of a mathematical model based simulation process thatfollows the procedures shown in the flow chart of Figure 6.5.

262 Terramechanics

Table 6.7. Specifications of test vehicle.

Vehicle weight W (kN) 40Track width B (cm) 180Contact length D (cm) 480Average contact pressure pm (kPa) 2.31Radius of front-idler Rf (cm) 25Radius of rear sprocket Rr (cm) 25Grouser height H (cm) 3Grouser pitch Gp 9Eccentricity of center of gravity e 0.00Height of center of gravity hg (cm) 95Distance of application point from axis lb (cm) 300Distance of application point from axis Ld (cm) 300Height of application point of force hb (cm) 50Height of application point of force hd (cm) 50Initial track belt tension H0 (kN) 29.4Circumferential speed of track (driving) V ′(cm/s) 100Vehicle speed (during braking state) V (cm/s) 100

The specifications of the vehicle system to be simulated are shown in Table 6.7. Theterrain-track system constants that develop between a track model plate equipped withstandard T shaped grousers and a shallow deposited snow covered terrain of depth 20 cmare k2 = 0.315 N/cmn2+1, n2 = 1.220, k4 = 32.34 N/cmn4+1, n4 = 0.862 from track modelplate loading and unloading test results, fs = 1.86, fm = 0.01 and jm = 1.5 cm for hump-typeshear deformation relations, and c0 = 0.685 cm2c1−c2+1/Nc1 , c1 = 0.694 and c2 = 0.476 forslip sinkage relations from track model plate traction test results.

(1) At driving stateThe variations of the driving force T1, the effective driving force T4, the amounts of sinkageof the front-idler s′

fi and the rear sprocket s′ri, the angle of inclination of the vehicle θ′

ti,the eccentricity of ground reaction ei, the various energy factors E1 ∼ E4 and the tractivepower efficiency Ed are a function of the slip ratio id can now be simulated.

The data are for an over snow vehicle working under driving action and running on a flatsnow covered terrain of β = 0 rad.

Figure 6.48 shows the results of the simulation in terms of the relations between T1,T4 and id . Both the values of T1 and T4 take maximum values of T1max = 69.6 kN andT4max = 68.6 kN at id = 0.3% respectively. Also, both these factors decrease rapidly aftertaking a peak value. Figure 6.49 shows the relations between the sinkages s′

fi, s′ri and id . The

front sinkage s′fi takes a minimum value of 3.1 cm at id = 0.3% and after that it increases

gradually. The rear sinkage s′ri increases parabolically with increasing values of slip ratio

id . The rear sinkage s′ri is always larger than the front sinkage s′

fi due to the increasingamount of slip sinkage that occur with id . As shown in Figure 6.50, the vehicle inclinationθ′

ti reaches a peak value of 0.011 rad at id = 0.5% and the eccentricity ei takes a maximumvalue of 0.0802 at id = 0.3%. After these points both these factors decrease rapidly.

Figure 6.51 shows the relations between the various energy components – namelythe effective input energy E1, the compaction energy E2, the slippage energy E3 andthe effective tractive effort energy E4 and the slip ratio id . The effective input energyE1 and the effective tractive effort energy E4 both have peak values at id = 0.3% and then


Figure 6.48. Relationship between driving force T1 effective driving force T4 and slip ratio id (drivingstate).


ri and slip ratioid (driving state).

decrease rapidly. E2 and E3 increase linearly with id . E4 has a peak value of 6843 kN cm/sat idopt = 0.3%. Figure 6.52 shows the relationship between the tractive power efficiencyEd and slip ratio id . Ed takes a maximum value of 98.6% at id = 0.05% and after that itdecreases gradually.

To summarise, when the simulated flexible tracked over-snow vehicle is running ona snow-covered terrain during driving action and is operating at idopt = 0.3% under the

264 Terramechanics


ei and slip ratio id (driving state).

Figure 6.51. Relationship between energy elements E1, E2, E3, E4 and slip ratio id (driving action).

maximum effective tractive effort energy, T1opt = 69.6 kN and T4opt = 68.6 kN can bedeveloped at s′

fi = 3.1 cm, s′ri = 8.3 cm, θ′

ti = 0.011 rad, ei = 0.0802 and Ed = 98.4%.The contact pressure distribution acting on the simulated flexible tracked over-snow

vehicle varies with the slip ratio id . Figure 6.53 shows the distributions of normal stress p′i(X )

and shear resistance τ ′i (X ) at id = 0.3, 0.6 and 0.9%. As shown in the diagrams, there are no

stress concentrations under the road rollers. This also shows that repetitive loading does notoccur on the snow-covered terrain due to the deflection of the track belt between road rollers.

The normal stress p′i(X ) tends to increase toward the rear part of the track belt due to

the positive value of ei. In spite of the large amount of deflection of the front part of the


Figure 6.52. Relationship between tractive power efficiency Ed and slip ratio id (driving state).

Figure 6.53. Contact pressure distribution (during driving action).

track belt due to the relatively small track belt tension, p′i(X ) does not show any major wavy

distribution. This means that there is no unloading process on the contact part of the trackbelt that accompanies increasing values of the angle of inclination of the vehicle. On theother hand, τ ′

i (X ) shows a gentle wavy distribution and a large thrust is developed at thefront part of the track belt.

Figure 6.54 shows the distributions of the track belt tension T0 around the track belt atid = 0.3, 0.6 and 0.9%. The belt tension T0 increases almost linearly toward the rear partof the track belt while the initial track belt tension of H0 = 29.4 kN acts on the base part ofthe front-idler.

At the base part of the rear sprocket, the track belt tension reaches 62.9 kN at id = 0.3%.The track belt at the base part of the rear sprocket is tensioned so as to provide a deflectionof 1.6 mm while the amount of deflection at the base-part of the front-idler is 3.2 mm forid = 0.3%.

266 Terramechanics

Figure 6.54. Distribution of track belt tension T0 (during driving action).

Figure 6.55. Relationship between braking force T1 effective braking force T4 and skid ib.

(2) At braking stateIn the same manner as has already been worked through, the variations of the braking forceT1, the effective braking force T4, the amounts of sinkage of the front-idler s′

fi and therear sprocket s′

ri, the angle of inclination of the vehicle θ′ti, the eccentricity of the ground

reaction ei, the various energy components E1 ∼ E4 and the braking power efficiency Eb

with variation in skid ib during braking action of the over snow vehicle running on a flatsnow covered terrain of β = 0 rad can be simulated.

Figure 6.55 shows the results of the simulation and plots the relations between T1,T4 for variation in ib. The forces T1 and T4 have peak values of T1max = 11.6 kN andT4max = 11.1 kN at ib = −0.3% respectively. After this peak they decrease rapidly to aminimum value of T1min = −27.4 kN and T4min = −29.2 kN at ib = −1.4% respectivelyand after that they increase gradually.

Figure 6.56 graphs the relations between s′fi, s′

ri and ib. The front sinkage s′fi decreases

gradually with ib after taking a maximum value of 8.4 cm at ib = −1.5%. The rear sinkages′

ri increases parabolically with increasing values of |ib|. The rear sinkage s′ri is always larger

than the front sinkage s′fi due to the increasing amount of slip sinkage that occurs at the rear

sprocket.Also, as shown in Figure 6.57, the inclination θ′

ti increases with increasing values of|ib| and the eccentricity ei takes a maximum value of 0.0138 at ib = −0.3%. After a peak



ri and skid ib

(braking state).


ei and skid ib (braking state).

ei moves rapidly to a minimum value of −0.0345 at ib = −1.4%. After this, it increasesgradually.

Figure 6.58 shows the relations between the effective input energy E1, the compactionenergy E2, the slippage energy E3, the effective braking force energy E4 and the skid ib.Both the value of E1 and E4 take peak values at the same skid of ib = −0.3%. After that,they both drop down to minimum values at ib = −1.4%. Then, they increase gradually. Theeffective input energy |E1| takes a maximum value of 2706 kNcm/s at ibopt = −1.4%. Thebraking power efficiency Eb takes a minimum value of 34.4% after that it increases almostlinearly with increasing values of |ib|.

To summarise, when the particular simulated flexible tracked over-snow vehicle is run-ning on a snow covered terrain during braking action and is operating at ibopt = −1.4% under

268 Terramechanics

Figure 6.58. Relationship between energy elements E1, E2, E3, E4 and skid ib (braking state).

Figure 6.59. Contact pressure distribution (during braking action).

maximum effective input energy, T1opt = −27.4 kN and T4opt = −29.2 kN can be developedat s′

fi = 8.4 cm, s′ri = 12.6 cm, θ′

ti = 0.009 rad, ei = −0.0345 and Eb = 108.0%. This showsthat a repetitive loading occurs on the snow-covered terrain.

The contact pressure distribution acting on a flexible tracked over snow vehicle varieswith skid ib. For example, Figure 6.59 shows the simulated distributions of normal stressp′

i(X ) and shear resistance τ ′i (X ) at ib = −1.4%. As shown in the diagram, there are stress

concentrations that occur just under the road rollers and p′i(X ) decreases between the road

rollers due to the deflection of the track belt. The normal stress p′i(X ) tends to increase

toward to the front part of the track belt due to a negative value of ei. At the rear partof the track belt, both the magnitude of the deflection of the track belt and the amplitude ofthe distribution of normal stress increase due to the relatively small value of the track belttension. On the other hand, τ ′

i (X ) shows the same wavy distribution, but the value of shearresistance changes from positive value to negative one at some point on the track belt. Afterthat, |τ ′

i (X )| increases toward to the rear part of the track belt and then decreases gradually.This phenomenon is a special feature of snow covered terrain and it is evident that the dragis developed mainly on the front side of the middle part of the track belt.


Figure 6.60. Distribution of track belt tension T0 (braking state).

Figure 6.60 shows the distributions of the track belt tension T0 around the track belt atib = −1, −2, and −3%. T0 increases toward to the front part of the track belt whilst theinitial track belt tension of H0 = 29.4 kN acts on the base part of the rear sprocket. At the basepart of the front-idler, the track belt tension reaches 50.0 kN at ib = −2%. The track belt atthe base part of the front-idler is tensioned as to provide an amount of deflection of 0.3 mmwhilst the amount of deflection at the base part of the rear sprocket is 2.55 mm at ib = −2%.

6.7 SUMMARY

In this chapter we have addressed one of the very difficult problems in everyday terrame-chanics, namely the problem of predicting the performance of a specific tracked-vehiclewith flexible track system which is required to operate upon a particular complex terrain.

In an attempt to solve this problem in a generic manner, a modeling system based ona mixture of experimentally defined parameters and analytical considerations has beendeveloped. By use of this modeling system and by means of a process of recursive analysis(using computers) it has been shown that the interplay between the many different designparameters and terrain characteristics can described. By comparison with experiments ithas been shown that this description is fairly good approximation of reality.

While the analyses of this chapter are at times complex and algebraically messy, theytruly address a very important industrial applications area. Most commercial bulldozers andtracked agricultural machines have flexible tracks and their field behaviour – if one includestrack static sinkage, slip sinkage and non-linear under-track soil failure – is very complex.Unfortunately, a complex multi-factorial theory is required to describe such complex realworld behaviour.

Finally, and in conclusion, this book has really been an attempt to mathematically-modelcomplex systems. To the extent that the reader considers that this aim has been achievedmethodologically and to the extent that the specific models developed are useful, this bookwill have achieved success.

REFERENCES

1. Wong, J.Y. (1989). Terramechanics and Off-Road Vehicles. pp. 105–137. Elsevier.2. Muro, T. (1989). Stress and Slippage Distributions under Track Belt Running on a Weak Terrain.

Soils and Foundations, Vol. 29, No. 3, pp. 115–126.3. Muro, T. (1989). Tractive Performance of a Bulldozer Running on Weak Terrain. J. of

Terramechanics, Vol. 26, No. 3/4, pp. 249–273.

270 Terramechanics

4. Muro, T. (1995). Trafficability Control System for a Tractor Travelling up and down a WeakSlope Terrain using Initial Track Belt Tension. Soils and Foundations, Vol. 35, No. 1, pp. 55–64.

5. Muro, T. (1991). Optimum Track Belt Tension and Height of Application Forces of a BulldozerRunning on Weak Terrain. J. of Terramechanics, Vol. 28, No. 2/3, pp. 243–268.

6. Karafiath, L.L. & Nowatzki, E.A. (1978). Soil Mechanics for Off-Road Vehicle Engineering.pp. 429–462. Trans Tech Publications.

7. Rowland, D. (1972). TrackedVehicle Ground Pressure and its Effect on Soft Ground Performance.Proc. 4th Int. Conf. ISTVS, Stockholm, Sweden, Vol. 1, pp. 353–384.

8. Cleare G.V. (1971). Some Factors which Influence the Choice and Design of High-Speed TrackLayers. J. of Terramechanics, Vol. 8, No. 2.

9. Fujii, H., Sawada, T. & Watanabe, T. (1984). Stresses in situ Generating by Bulldozers. Proc. 8thInt. Conf. ISTVS, Cambridge, England, Vol. 1, pp. 259–276.

10. Sofiyan, A.P. & Maximenko, Ye.I. (1965). The Distribution of Pressure under a TracklayingVehicle. J. of Terramechanics, Vol. 2, No. 3, p. 11.

11. Yong, R.N. & Fattah, E.A. (1975). Influence of Contact Characteristics on Energy Transfer andWheel Performance on Soft Soil. Proc. 5th Int. Conf. ISTVS, Detroit, U.S.A., Vol. 2 pp. 291–310.

12. Sugiyama, N. (1976). Traffic Performance of a Crawler. Construction Machinery and Soil.pp. 56–63. Japan Industrial Press. (In Japanese).

13. Hata, S. (1987). Theory of Construction Machinery. pp. 91–93. Kashima Press. (In Japanese).14. Bekker, M.G. (1956). Theory of Land Locomotion. pp. 186–244. The University of Michigan

Press.15. Bekker, M.G. (1960). Theory of Land Locomotion. pp. 101–112. The University of Michigan

Press.16. Wills, B.M.D. (1963). The Measurement of Soil Shear Strength and Deformation Moduli and

a Comparison of the Actual and Theoretical Performance of a Family of Rigid Tracks. J. ofAgricultural Engineering Research, Vol. 8, No. 2.

17. Torii, T. (1976). Trafficability of Construction Machinery. Construction Machinery and Soil.pp. 47–55. Japan Industrial Press. (In Japanese).

18. Wong, J.Y. (1989). Terramechanics and Off-Road Vehicles. pp. 121–137. Elsevier.19. Ito, G., Maeda, T. & Ohta, H. (1983). Contact Pressure Distribution of a Tracked Vehicle. Proc.

of Symp. on Construction Machinery and Works. pp. 17–20.20. Bekker, M.G. (1969). Introduction to Terrain-Vehicle Systems. pp. 482–491. The University of

Michigan Press.21. Yong, R.N., Elmamlouk, H. & Della-moretta, L. (1980). Evaluation and Prediction of Energy

Losses in Track-Terrain Interaction, J. of Terramechanics, Vol. 17, No. 2, pp. 79–100.22. Yong, R.N., Fattah, E.A. & Skiadas, N. (1984).VehicleTraction Mechanics. pp. 195–255. Elsevier.23. Garber, M. & Wong, J.Y. (1981). Prediction of Ground Pressure Distribution under Tracked

Vehicles, J. Terramechanics, Vol. 18, No. 1, pp. 1–23.24. Wong, J.Y. (1986). Computer aided Analysis of the Effects of Design Parameters on the

Performance of Tracked Vehicles. J. Terramechanics, Vol. 23, No. 2, pp. 95–124.25. Oida, A. (1976). Analysis of Tractive Performance of Track-Laying Tractors, J. of Agricultural

Machinery, Vol. 38, No. 1, pp. 25–40. (In Japanese).26. Muro, T. (1990). Automated Tension Control System of Track Belt for Bulldozing Operation.

Proc. of the 7th Int. Symp. on Automation and Robotics in Construction, Bristol Polytechnic,Bristol, England. pp. 415–422.

27. Muro, T. (1991). Tension Control System of Track belt of a Tractor Carrying Down Weak Slopeat Braking State. Proc. 2nd Symp. on Construction Robotics in Japan. pp. 41–50, JSCE et al.(In Japanese).

28. Muro, T. (1991). Initial Track Belt Tension Affecting on the Tractive Performance of a Bull-dozer Running on Weak Terrain. Terramechanics, Vol. 11, pp. 15–20. The Japanese Society forTerramechanics. (In Japanese).


EXERCISES

(1) Show that the effective input energy per second E1 of a flexible tracked vehicle duringdriving action can be expressed as the sum of the compaction energy E2, the slippageenergy E3, the effective driving force energy E4, and the potential energy E5.

(2) Also, show that the effective input energy per second E1 of a flexible tracked vehicleduring braking action can be expressed as the sum of the compaction energy E2, theslippage energy E3, the effective braking force energy E4, and the potential energy E5.

(3) Compare the relationships between the amount of sinkage of the front-idler and therear sprocket and slip ratio which are shown in Figure 6.13 for the driving state andin Figure 6.20 for the braking state. Explain the fact that, for the same absolute valueof slip ratio and skid i.e. id = |ib|, the amount of sinkage of the rear sprocket duringdriving action is always larger than that which occurs during braking action.

(4) The distribution of the normal contact pressure p′i(X ) on a flexible tracked vehicle

is shown in Figure 6.17(a) for the driving state and in Figure 6.23(a) for the brakingstate. During driving action, the amplitude of the wavy distribution tends to increasetoward the forward part of the track belt, while, during braking action, the amplitudetends to increase toward the rear part. Discuss the reasons for these phenomena.

(5) As mentioned in Section 6.6.1 (2), a bulldozer operating on a soft silty loam terraincan be affected to a very large extent by the size of the vehicle. Calculate the maximumdrawbar pull T4max for a bulldozer of weights W = 12.5 kN and 200 kN respectivelythrough use of the terrain-track system constants given in Table 6.3.

(6) The effect of the initial track belt tension H0 on the tractive power efficiency Ed ofa bulldozer operating on a soft silty loam terrain has been presented in Section 6.6.1(3). The dimensions of a particular bulldozer are given in Table 6.5. Calculate the ratioof variation of the tractive power efficiency Ed for H0 = 0 kN to 50 kN, for the casewhere the amount of eccentricity of the center of gravity of the vehicle is e = 0.00.

(7) An analytical study of the tractive performances of a small bulldozer operating ona decomposed granite soil has been given in Section 6.6.2 (1). The dimensions of thebulldozer are shown in Table 6.6. Describe in detail the tractive performance of thebulldozer when it is operating under maximum effective tractive effort energy.

(8) The analytical braking performance of a small bulldozer operating on a decomposedgranite soil was modeled in Section 6.6.2 (2). The dimensions of the bulldozer areshown in Table 6.6. Describe in detail its braking performance when it is operatingunder maximum input energy.

(9) An analytical study of the tractive performance of an over snow tracked vehicle ofweight 40 kN operating on a shallow snow covered terrain of depth 20 cm has beengiven in Section 6.6.3 (1). Describe in detail the tractive performance of the over snowvehicle when it is operating under the condition of maximum drawbar pull energy.

(10) Some analytical results of the braking performances of an over snow tracked vehicleoperating on a snow covered terrain have been presented in Section 6.6.3 (2). The vehi-cle dimensions are as given in Table 6.7. Describe in detail the braking performance ofthe over snow vehicle when it is operating under conditions of maximum input energy.

Index

A type function, exponential function, 125air pressure, critical, 90angle of internal friction, 5, 16angle of lateral slippage

for cornering flexible tires, 112angle of lateral slippage, 112apparent effective braking force

for braked flexible tired wheel on a slope,107, 122

asphalt pavementtrack plate loading tests on, 132

average contact pressure, 90versus tire air pressure, 97

B Type function, hump function, 125bead, of tire, 83bearing capacity, 6, 10bearing capacity of terrain, 10, 134Bevameter, 12bias tires, 84box shear test, 8braked flexible tired wheel

sinkage distribution under, 105braked flexible tired wheel,

slippage distribution under, 105braking efficiency, 109braking force

re braking rigid wheel, 59, 61effective, 59, 67for rigid track vehicle & braking action, 197

braking performance,of experimental rigid wheel systems, 68, 74of simulated rigid wheel systems, 68, 74

braking workbraked flexible tired wheel on a slope, 108

Buckingham-Pi principles, 135bulldozing effect, 8, 10bulldozing resistance

and model-track-plate test, 142, 148bulldozing zone, under rigid wheel, 35

calorimeter, 20, 33carcass, of tire, 85casing, of tire, 85CI (cone index), 18clay, 5coefficient of braking resistance, 93, 111, 121

for flexible tire wheel on hard terrain, 93of braking rigid wheel, 60

coefficient of cornering, 116

coefficient of curvature, 4coefficient of distortion, 136coefficient of driving resistance, 92, 110, 121

for flexible tire wheel on hard terrain, 92coefficient of earth pressure

re model-track plate 127coefficient of estimation, 136coefficient of fillness, 88coefficient of friction

for cornering flexible tires, 113coefficient of propagation, 25coefficient of rolling friction, 92, 110coefficient of rolling resistance,

for a rolling rigid wheel, 60coefficient of subgrade reaction, 6coefficient of traction

for flexible tire wheel on soft terrain, 94re model-track plate, 128

coefficient of uniformity, 4coefficients of bearing capacity, 149coefficients of deformation, 126, 136, 138coefficients of sinkage, 124cohesion, 6, 7, 9, 18compaction effects

under flexible tired wheel, 96compaction energy, 35

braked flexible tired wheel on a slope, 108driven flexible tired wheel on a slope, 103

compaction resistancebraked flexible tired wheel on a slope, 107of braking rigid wheel, 65of driven rigid wheel, 51under rigid track vehicle, 167, 182for rigid track vehicle – soft terrain,

167, 182computer simulation of tractive performance

rigid track vehicle during driving action, 172computerised simulation

experimental validation of, 236computerised simulation of traffic performance

rigid track vehicle during braking, 184, 197flexible track vehicle driving & braking, 184

concrete pavementtraversal by rigid track vehicle, 197

cone index (CI), 18Cone penetration test, 17cone penetrometer, 18cone plate, 13consistency, 6consistency index, 6

274 Index

consolidated undrained shear test, 17contact pressure

under pneumatic tire on loamy terrain, 116under a braking rigid wheel, 61

cornering characteristics,flexible tired wheels, 109

cornering power, 116Coulomb’s failure criterion, 9critical average contact pressure, 97critical density, snow, 29crown, of tire, 85CU test, 17cycloid curve, 43

D.B.P., 204, 205D test, 17decomposed granite sandy terrain

plate traction tests on, 130traversal by flexible track vehicle, 254

deformation energy, of flexible tire, 90density, threshold, 25dilatation, 126dimensional analysis, 135direct shear test, 8distorted model, 136drag, for rigid track vehicle

during braking action, 175, 176drainage shear test (D test), 17driving force, for rigid track vehicle

during driving action, 197driving force energy,

for driven flexible wheel on slope, 103dry density, 4, 20Dunlop, John, 83dynamic box shear test, 10

effective braking force energyre braking rigid wheel, 67

effective braking force,for braking rigid track vehicle, 184

effective drawbar pull energy,for driven rigid wheel, 54

effective driving force,for driven rigid track vehicle, 171of driven rigid wheel, 54

elliptical curves,for cornering flexible tires, 113

energy equilibrium analysis,for driven rigid wheel, 54for braking rigid wheel, 67for driven rigid track vehicle, 170for braking rigid track vehicle, 184for flexible track vehicles, 209, 212

exponential type function, B Type function, 125

F function, 7F.E.M., 13

re flexible tires, 83under rigid wheel, 35

flexible tired wheels, 83action on rigid terrains, 85braking action on hard terrains, 91compaction effects of, 96contact pressure distribution under, 86cornering characteristics, 112critical air pressure, 90deformation energy of, 90deformation shape of, 96deformation under static load, 87distribution of ground reaction under, 98distribution of slippage under, 99driving action on hard terrains, 9driving torque, 92, 93elastic modulus effects, 88forces acting on while cornering, 112lateral rigidity of, 89mechanical characteristics of, 86on hard terrains, 83on soft terrains, 94operating on slopes, 101rolling action on hard terrains, 91shape of contact part in cornering, 115slippage under braking action, 91static load bearing capacity, 85structure of, 84terrain deformation under, 96trafficability on soft terrain, 83, 94

flexible tired wheels, 83flexible track belt, 123, 209flexible track vehicle, 209

energy equilibrium analysis of, 209force balance analysis of, 209deformation of track belt, 212simulation of traffic performance, 215–224theory of steering motion of, 224thrust and steering ratio, 228slippage in turning motion, 229turning resistance moment, 231computer simulation traffic performance, 232steering performance of, 233contact pressure under, 236, 238soil particle behaviour under, 237traffic performance over

silty loam terrain, 239–253decomposed granite sandy terrain, 254snow covered terrain, 261

flexible track vehicles, 209flexible vehicle with rubber grousers

terrain-track system constantsfor silty loam terrain, 250

Index 275

flexible wheel, flexible to rigid transition, 81, 83floatation, 1force balance, of braking rigid wheel, 60front idler sinkage

rigid track vehicle, 160rigid track vehicle – braking action, 180, 195

front idler slippage, for rigid track vehicleduring braking action, 180

good distribution, 5grain size analysis, 4gravel, 5grouser, 13

height, 124pitch, 124pitch-height ratio, 127

Hopkinson’s bar, 10hump type

behaviour, 9function, A type function, 125

hydrometer, 4

ice, 19indices of sinkage, 124

joint-type calorimeter, 20

kinematicsof braking motion, 111of driving motion, 110, 111of rolling motion, 110

Kinoshita’s hardness meter, 21

land locomotion resistance, 46latent heat, 20lateral coefficient of friction

for cornering flexible tires, 113lateral slip ratio

for cornering flexible tires, 113liquid limit, 6loading rate, 10longitudinal coefficient of friction

for cornering flexible tires, 112longitudinal skid

for cornering flexible tires, 112longitudinal slip ratio

for cornering flexible tires,

mobility, 1, 2mobility index (MI), 18model-track-plate loading test

scale effects on, 134model-track-plate loading test

see also plate loading test, 123

model-track-vehiclerunning on remoulded silty loam, 128

modified roundness, 5modulus of deformation, 11Mohr-Coulomb failure criterion, 15

nylon cord tires, 85

off-the-road tires, functional needs of, 84optimum skid

for braked flexible tired wheel on slope, 109for braking rigid track vehicle, 184

optimuum effective driving forcefor driven flexible tired wheel on a slope, 104

over-snow vehicle, 204, 261

penetration depth, 24, 25, 26penetration pressure, 18, 22, 23, 25Pi terms, 135plastic limit, 6plasticity, 6plate loading test, 6, 14

snow, 23ply, tire, 83pore water, 3porosity, 3potential energy

braked flexible tired wheel on a slope, 108driven flexible tired wheel on a slope, 103

pressure bulb, model-track-plate test, 140, 142pressure sinkage curve, 13prototype, 137, 138pycnometer, 3

radial tires, 85Rammsonde, 21rating cone index (RCI), 18ratio

of anisotropy, 13of similarity, 122, 123

rear idler sinkage, rigid track vehicle, 159rear sprocket sinkage

rigid track vehicle during braking, 179, 198rectangular plate loading test, snow, 23, 24relative density, 4remoulding, 18

index (RI), 19rigid tire, 90rigid track belt, 123rigid track vehicle, 149

compaction resistance of, 167computer simulation of tractive performance

during driving action, 172effective driving force, 171energy analysis during driving action, 170

276 Index

rigid track vehicle contd.system of forces on driven machine, 159pressure under driven machine, 160pressure under stationary machine, 149sinkage under stationary machine, 149slippage and driving action, 158thrust analysis on, 162

rigid track vehiclesystem of forces on braking machine, 174energy analysis during braking action, 183effective braking force, 184computer simulation of traffic performance

during braking action, 184, 185, 197experimental trials of, 187–197traffic performance on concrete, 197, 202performance on snow covered terrain, 202

rigid wheel, compaction resistance, 51contact pressure under, 35contact pressure under wheel at rest, 36driving force, 47effective driving force, 53energy balance of, 54flexible to rigid transition, 96, 98forces acting on driven wheel, 45sinkage under wheel at rest, 36slippage under driving action, 39static bearing capacity, 36

rigid wheels, 35pressure under rolling, 35

ring loading plate, 13ring shear test apparatus, 30road rollers, in undercarriage systems, 123road rollers, on flexible track vehicles, 209rolling frictional resistance,

of flexible track vehicle, 237running gear, 1rut

under rigid track vehicle, 154depth

under flexible tired wheels, 102, 108, 121

sand, 5saturated soil, 3saturation, 3, 21sedimentation analysis, 4self aligning torque,

for cornering flexible tires, 114sensitive clay, 12sensitivity ratio, 12, 18shape coefficient, 14shape coefficients,149shear box, 8shear deformation, 6, 8, 10, 13, 19shear resistance, 8, 9, 26shoulder, of tire, 85

shrinkage limit, 6side wall, of tire, 84sieve analysis, 4silt, 5silty loam terrain

flexible track vehicle over, 239–253similitude, 136, 146simulation analysis

of performance of rigid wheels, 68simulation, experimental validation of, 184simulation modelling of traffic performance

for flexible track vehicleduring driving and braking, 215–223

sinkage, 6, 7, 8, 13sinkage deformation energy,

re braking rigid wheel, 67re driven rigid wheel, 54

sinkage distributionunder a braked flexible tired wheel, 105

sinkage, static, 6, 7sintered snow, 19, 25, 27size effect, 123, 134, 135size effect of vehicle, 246skid, of braking rigid wheel, 55slenderness ratio, 5slip, 8

under rigid wheel, 35under track belt, 127

slippage, under braking rigid track vehicle, 173slip sinkage, 7, 8, 10, 13, 15

for braked flexible tired wheel on slope, 107of model-track-plate test, 142under track belt, 124, 126

slip velocity, 10, 155slippage

of braking rigid wheel, 55under driven rigid track vehicle, 155of driven rigid wheel, 39

slippage distributionunder a braked flexible tired wheel, 99, 105

slippage energy, 35braked flexible tired wheel on a slope, 108driven flexible tired wheel on a slope, 103re braking rigid wheel, 67re driven rigid wheel, 54

sloped terrains, and flexible tire action, 101snow wet, 19snow covered terrain, 19, 144

and model-track-plate test, 142shear resistance of, 145slippage of model-track-plate, 145traction tests on, 142traversal by rigid track vehicle, 204flexible track vehicle performance on, 261

snow density, 20

Index 277

snowcompressive strength of, 24deformation characteristics of, 24elastic modulus, 24fresh, 19hardness of, 21shear tests on, 26sintered, 19vane cone tests on, 26

snowmobiles, 123soil deformation

under a towed rigid wheel, 56, 57under driven rigid wheel, 40

specific gravity, 3

terrainelastic modulus of, 88trafficability, 18

terrain-track system constants, 123for bulldozer, 239for decomposed granite sandy terrain, 132for silty loam terrain, 129for snowy terrain, 144rubber grousers, for silty loam terrain,

252terrain-wheel system constants

for experimental system, 68threshold density, 29tire

aspect ratio, 85cornering characteristics of, 112, 116

tire deformation shape,for braked flexible tired wheel, 98

track beltmulti-roller type, 123shear resistance under, 124slip sinkage under, 131slippage under, 124structures, 123deformation shape of flexible belt, 212

track belt tensionfor flexible track vehicle, 246, 249, 252

track plate loading test, 124track plate traction test, 124track plate unloading test, 124track-laying vehicles, 123tractive efficiency

for flexible tire wheel on soft terrain, 94

traction workfor driven flexible tired wheel on a slope, 103

tractive performanceof experimental rigid wheel systems, 68of simulated rigid wheel systems, 68

trafficability, 1, 2, 6, 11, 18, 21of rigid wheel, 35

treadelastic modulus of, 88of tire, 75block type, 86smooth type, 86traction type, 86rib type, 86rock type, 86

tread pattern, effect on driving force, 95tread patterns, for off road tires, 86Tresca’s failure criterion, 15triaxial compression test, 15trochoid curve

of braking rigid wheel, 58of rigid wheel, 43, 82

Type A soil, 9Type B soil, 9tyre – see tire

unconfined compression test, 10unconfined compressive strength, 11unconsolidated undrained shear test, 17undrained shear strength, 11, 17unit weight, 3unsaturated soil, 3untraction state, 106, 136UU test, 17

vane cone, snow, 26vane shear test, 12

for snow, 26vehicle cone index (VCI), 18void ratio, 3Von Mises’s failure criterion, 15

water content, 3weak terrain, bearing capacity of, 35wet density, 3wheel mobility numbers, 94, 95wheel, rigid, rolling locus of, 42wheel, rigid – see rigid wheel, 35

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Terramechanics